How to find the joint speed. Tasks for movement to prepare for the exam in mathematics (2020). Solving problems for independent work

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Beginning in 5th grade, students often come across these problems. Also in primary school students are given the concept of "general speed". As a result, they form not entirely correct ideas about the speed of approach and the speed of removal (there is no such terminology in elementary school). Most often, when solving a problem, students find the sum. It is best to start solving these problems with the introduction of the concepts: “rapprochement rate”, “removal rate”. For clarity, you can use the movement of the hands, explaining that bodies can move in one direction and in different directions. In both cases, there may be an approach speed and a removal speed, but in different cases they are found in different ways. After that, students write down the following table:

Table 1.

Methods for finding the speed of approach and speed of removal

When analyzing the problem, the following questions are given.

Using the movement of the hands, we find out how the bodies move relative to each other (in one direction, in different ones).

We find out what action is the speed (addition, subtraction)

We determine what speed it is (approach, removal). Write down the solution to the problem.

Example #1. From the cities A and B, the distance between which is 600 km, at the same time, a truck and a car left towards each other. The speed of the passenger car is 100 km/h, and the speed of the truck is 50 km/h. In how many hours will they meet?

Students use their hands to show how cars move and draw the following conclusions:

cars move in different directions;

the speed will be found by addition;

since they are moving towards each other, then this is the speed of convergence.

100+50=150 (km/h) – closing speed.

600:150=4 (h) - the time of movement before the meeting.

Answer: after 4 hours

Example #2. The man and the boy left the state farm for the garden at the same time and go the same way. The man's speed is 5 km/h and the boy's speed is 3 km/h. How far apart will they be after 3 hours?

With the help of hand movements, we find out:

the boy and the man are moving in the same direction;

speed is the difference;

the man walks faster, i.e., moves away from the boy (removal speed).

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2. SPEED OF THE BODY. RECTILINEAR UNIFORM MOVEMENT.

Speed is a quantitative characteristic of the movement of the body.

average speed- This physical quantity, equal to the ratio of the point displacement vector to the time interval Δt, during which this displacement occurred. The direction of the average velocity vector coincides with the direction of the displacement vector . average speed is determined by the formula:

Instant Speed, that is, the speed in this moment time is a physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval Δt:

In other words, instantaneous speed at a given point in time is the ratio of a very small movement to a very small period of time during which this movement occurred.

The instantaneous velocity vector is directed tangentially to the trajectory of the body (Fig. 1.6).

Rice. 1.6. Instantaneous velocity vector.

In the SI system, speed is measured in meters per second, that is, the unit of speed is considered to be the speed of such uniform rectilinear motion, in which in one second the body travels a distance of one meter. The unit of speed is denoted m/s. Often speed is measured in other units. For example, when measuring the speed of a car, train, etc. The commonly used unit of measure is kilometers per hour:

1 km/h = 1000 m / 3600 s = 1 m / 3.6 s

1 m/s = 3600 km / 1000 h = 3.6 km/h

Addition of speeds (perhaps not necessarily the same question will be in 5).

The velocities of the body in different reference systems are connected by the classical law of addition of speeds.

body speed relative to fixed frame of reference is equal to the sum of the velocities of the body in moving frame of reference and the most mobile frame of reference relative to the fixed one.

For example, a passenger train is moving along a railroad at a speed of 60 km/h. A person is walking along the carriage of this train at a speed of 5 km/h. If we consider the railway to be motionless and take it as a frame of reference, then the speed of a person relative to the frame of reference (that is, relative to railway), will be equal to the addition of the speeds of the train and the person, that is

60 + 5 = 65 if the person is walking in the same direction as the train

60 - 5 = 55 if the person and the train are moving in different directions

However, this is only true if the person and the train are moving along the same line. If a person moves at an angle, then this angle will have to be taken into account, remembering that speed is vector quantity.

An example is highlighted in red + The law of displacement addition (I think this does not need to be taught, but for general development you can read it)

Now let's look at the example described above in more detail - with details and pictures.

So, in our case, the railway is fixed frame of reference. The train that is moving along this road is moving frame of reference. The car on which the person is walking is part of the train.

The speed of a person relative to the car (relative to the moving frame of reference) is 5 km/h. Let's call it C.

The speed of the train (and hence the wagon) relative to a fixed frame of reference (that is, relative to the railway) is 60 km/h. Let's denote it with the letter B. In other words, the speed of the train is the speed of the moving reference frame relative to the fixed frame of reference.

The speed of a person relative to the railway (relative to a fixed frame of reference) is still unknown to us. Let's denote it with a letter.

We will associate the XOY coordinate system with the fixed reference system (Fig. 1.7), and the X P O P Y P coordinate system with the moving reference system. Now let's try to find the speed of a person relative to the fixed reference system, that is, relative to the railway.

For a short period of time Δt, the following events occur:

Then for this period of time the movement of a person relative to the railway:

This displacement addition law. In our example, the movement of a person relative to the railway is equal to the sum of the movements of a person relative to the wagon and the wagon relative to the railway.

Rice. 1.7. The law of addition of displacements.

The law of addition of displacements can be written as follows:

= ∆ H ∆t + ∆ B ∆t

The speed of a person relative to the railroad is:

The speed of a person relative to the car:

Δ H \u003d H / Δt

The speed of the car relative to the railway:

Therefore, the speed of a person relative to the railway will be equal to:

This is the lawspeed addition:

Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).

Rectilinear motion- this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.

Uniform rectilinear motion is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.

The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:

Speed ​​of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:

Thus, the speed of uniform rectilinear motion shows what movement a material point makes per unit of time.

moving with uniform rectilinear motion is determined by the formula:

Distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity on the OX axis is equal to the velocity and is positive:

v x = v, i.e. v > 0

The projection of displacement onto the OX axis is equal to:

s \u003d vt \u003d x - x 0

where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)

Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:

If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body velocity on the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид.

– Is it worth it to continue the relationship if you and your partner have different speeds of movement?

We sit in one of the small hotels in Nepal and traditionally act out the question. This is the last day in the mountains and the last time we pull anonymous notes. We are 14 people from different countries and cities, we have just completed a trek to the Langtang valley and to Gosaikunda lake.

Even at the start, in Kathmandu, all the participants of the track chipped in on an anonymous question. I, the presenter, took out one every evening and read the next problem aloud, which gave rise to a discussion, and sometimes disputes - through the prism of different experience, understanding of the situation, well, or delusions - a matter of life.

Our last night in the mountains. I once again unfold the paper, read first to myself, and then for everyone:

“Is it worth continuing the relationship if you and your partner have different speeds of movement?”

You can already hear the sound of air being drawn into the lungs. For three years of holding such conversations, the statistics were unchanged - questions about relationships were always the most popular. The group was preparing for a lively discussion.

But everyone was outstripped by that special quiet and calm timbre of the voice, which happens only in a person who does not need to prove anything:

“My thirty years of experience in marriage suggests that it is impossible to always have the same speed of movement with your partner,” said Olga, one of the participants in our trip. And she continued:

One way or another, there will be moments when one will be faster and the other slower. And a situation will inevitably come when they will change places, of course, if we talk about relations on long distance.

True, I didn’t hear anything anymore - like other opinions, if there were any that evening. Once every couple of years, if I'm lucky, life brings me to a phrase-book that endlessly unfolds its meaning. Once such a phrase was accidentally seen somewhere: "It is impossible to find oneself, one can only create oneself." Words that not only stunned me to the core, but literally turned my whole life upside down. That evening was special. I came across another phrase-book that could be read endlessly:

It is impossible to always have the same speed with your partner over a long distance.

For a long time I revolved around these words, trying to expand their meaning. I felt the truth behind them. But if with other phrases it was enough to push off a little, as I was ready to write a whole book, then here it didn’t go beyond a pleasant tickle, which is the essence of it. The texture of my own experience was missing. Then I came to Olga with a request to "beat off the pitch." Answer my questions that arise around the yes about this topic.

Olga responded with ease.

About different speeds of movement of partners and relationships over a long distance

Serves - Olesya Vlasova, author of the Re-Self blog. Married 9 months (in a relationship - 3 years). Beats - Olga Vakhrusheva, business consultant, married 32 years. When we met, Olga was 15, and Nikolai was 18 years old. They got married as soon as Olga turned 18. For 22 years they have been living in New Zealand, where they moved from Novosibirsk. Olga and Nikolai have two children and two grandchildren.

What about the one who is faster? From the outside, the story that in relationships over a long distance cannot always have the same speed for both partners sounds beautiful, and most importantly, one feels that there is truth behind these words, but from the inside, everything is not so simple and obvious. What about the one who is ahead today? Help the other? Or vice versa - leave him alone and not "drag on yourself"? And how to find peace in such a situation?

- For me, the statement that in a relationship over a long distance cannot always be the same speed for both partners, is an axiom. As well as the fact that two people building relationships are a priori different, two independent, unique personalities. Both are not ideal. But it's clear to me now.

When I was younger, I tried to build our intra-family relationships based on previously unviable attitudes: we must always do everything together and in full mutual understanding, we must be one, love is a gift that happens to you, which you find if you're lucky .

In practice, it turned out, of course, not so. And attempts to tie reality to a far-fetched ideal caused both misunderstandings, and insults, and quarrels that could have been avoided if the original views of the world were more viable.

I don’t know what is happening in young minds now and what ideas your generation grew up on, but in our time, girls from early childhood have seen and heard something like the following:

• In fairy tales and in films: a prince on a white horse will surely ride to the princess, he will love her more than life, they will always live happily, and he will solve all her problems.
• From the conversations of older women: a real man should ... And further down the list: earn, provide, be a support, be smart, caring, an excellent father, a loving husband, gentle, understanding, and so on. (in fact, many of these definitions are mutually exclusive).
• From the same source: real men died out in the world. You can't count on them. Either drunkards, or lazy and henpecked, or heartless careerists. You need to keep everything under control and, in fact, you can trust a man with caution.

So my head is full of ideas. There is only hope that the ideal relationship will happen by itself or he will make you happy. But now it is clear that no one else can make another person happy (no matter how hard he tries). This is an internal process that goes in parallel with the steps towards each other.

Back to your main question. What to do to the one who is faster today? The answer is I don't know. There is no universal answer for everyone. Sometime you need to help, sometime you need to leave it alone, sometime you need to give a guiding kick (with love). Often you just need to go about your business, do not panic, but make it clear that you are here, you are there and you care and love. If we are talking about two adequate people, and not about pathology, then simply understanding that this is not forever usually helps a lot.

In addition, a decrease in speed often has objective reasons:

• The difference in temperaments (you must learn to live with this if you want to maintain a relationship).
• Health problems that a man often does not talk about, and a woman invents God knows what.
• Problems at work or in business (which he also most often does not talk about until he figure out what to do about it).
• Some big changes to be aware of before taking the next step.
• The difference in age (and, accordingly, in speeds).
• Hormonal changes.
• Lastly, fear. Which men have no less, and maybe more than ours, but there is no one to turn to for help.

And here we are with our speeds and personal growth. In general, as my experience shows, this question often arises among young girls.

Let's talk about a young girl. She believes (objectively or not, another question), at least it seems to her that she is doing more - pulling work, children, home. But he doesn't. Does not help. Does less.

– Yes, it is familiar. It looks like he owes me. I earn, and even the children on me. Claims. Expectations. After three years of marriage, life begins - socks in the hallway, he didn’t say something, he didn’t do it.

We need to figure out the reasons. Analyze. Is it a temporary decrease in speed or is this the nature of lying on the couch? The second is unlikely to be close to a girl active in life. But there may be other reasons as well. Very often, we ourselves do not give our men a chance to get involved in the process.

For example, we voiced the problem (and often we didn’t voice it at all, but we hope that he will guess it himself). He has not yet had time to comprehend the problem, and we are already rushing to do and solve everything ourselves. Well, why would he then run a race with us? Or - why did you tell him about the problem then?

Or he did something, and we are unhappy - he didn’t do it right. Well, once it’s wrong, the second time it’s wrong, and then you don’t want to move (would you like to?). And why not put the question in a different way: “This is my area of ​​responsibility, and this is yours. How and what you do is your decision, but the result is expected to be such and such.” He may stumble once, maybe he will forget sometime, and then he will figure it out. If we believe that he will figure it out, and do not snort at every occasion.

This applies to everything. Starting with the elementary: instead of stating with resentment in his voice that he never takes out the trash, and you are all on your own, yourself ... But you also get tired ... and further in the text. It’s more productive to say: “Darling, let’s do this: taking out the garbage in the house is on you. I'm counting on you." And that's it. And forget. And don't take it out. And don't remind me. Even if the house starts to smell. He, too, will feel it, and will remember, and throw it away, and will already remember.

It is also very important to set specific goals for your partner and clearly and clearly ask for what we need. What are we looking for help with? Many things they simply do not see. They are not even aware of their existence at first. And our minds cannot be read. It's much easier to say, "Honey, I'm sewing up in the kitchen, please hang up the laundry and put the kids to bed." If a man is adequate and not busy at this moment with something important, then the issue is resolved. And what does a young woman usually do? He rushes about between the kitchen, laundry and children, waiting for him to understand (this is obvious), satane, offended. And you could just say.

The same rules apply to your relationship with your son. Apparently, boys perceive such language better.

And it is important to be aware of this simple thing that if at the moment a woman (or man) is stronger in a relationship, this does not mean that she (he) is always right (right).

– And about those who become weaker at some point and can reflect it? After all, it is also hard. A man, of course, but a girl capable of introspection will also feel uncomfortable: for some reason she is not in a rut, maybe pregnancy, maybe, I don’t know, illness or something, but he has a career, rise, development, movement. This is jealousy, and anxiety, and just a feeling of worthlessness can come out. Have you had it?

- Yes, just when moving to New Zealand. From the very beginning, we relied on my husband. He had a language, and he immediately went to study and work. He came home tired, but on the rise and with a bunch interesting information, acquaintances, plans. And I felt completely lost. I couldn’t do the simplest things myself (I don’t have a language, I don’t drive a car, I don’t know how the bank works, I don’t know anyone, my husband can’t provide support - he’s not at home all day, he has two small children in his arms). And a month ago, I owned businesses, advised people, taught, taught others what to do and how to do it.

It helped to realize that this was happening to me. That is, it is important not to deceive yourself and not to look for someone to blame, but to describe the situation in which I am currently in with maximum honesty.

• What's happening? Where am I now?
• Is this a temporary inconvenience or a real problem?
• How did I get here?
• What does not suit me in the situation?
• What can I do to change the situation?
• Map out real steps.
• Take these steps.
• Check the result with the planned, make corrections, if necessary.
• Move on.

In principle, I solve all my problems according to this algorithm. The most difficult thing is usually to become aware of your emotions, take yourself emotionally out of the situation and turn on your head. Sometimes I give myself permission for another week to “be hysterical and feel sorry for myself,” and then get down to business. Usually works.

Trying to ignore your emotions and fears doesn't exactly work. It’s easier for me to say to myself: “OK, I’m afraid of this scenario. Fine. Hello fear. Then ask yourself the question: “What will happen in the worst case if the fears come true? Is it deadly? What would option B be? Can I live with this? Most often, the answer is that you can live with it and it’s not so scary in reality. And then there is energy to look for options and move on.

The first months in New Zealand were painful for the complete zeroing, the loss of social contacts, status, skills, understanding of how to earn money, how life and society work, the transformation from a sociable professional into a silent “nothing”. But there were children in her arms, so it was impossible to fall into a complete hysteria. Therefore, a month later I went to learn the language (how - a separate detective story). Six months later, she went to work as a volunteer in a bureau to support poor families (she overcame the fear of communication, gained local experience, acquaintances), and after another six months she went to work in her specialty. Well, go ahead.

What is the most important thing in a long-term relationship?

- From what I saw in my life, from communication with couples who have lived a long life together and are happy together (and there are plenty of them, by the way, but somehow very little is said about this in modern media, more and more about problems ), - a simple trend emerges very clearly in the relationship of these couples.

All happy couples have mutual trust. I have not seen a single couple so that people do not trust each other and live happily ever after. It is impossible to live with a person and constantly expect a catch. It is a life of endless fear and stress. For both.

I also know couples where everything is not easy. Distrust fills their world. From the side it is clear that the most incredulous person usually has big problems with self-esteem, and besides, he (she) is sinful precisely in what he suspects his half of, or was very bad life experience, or expectations are very unrealistic.

That is, again we return to the question of our own fears, unrealistic expectations and other cockroaches in my head. The partner most often has nothing to do with it at all. You need to deal with yourself. In certain cases, you probably need to contact a specialist who can help specific people in a specific situation.

- And how to gain it, basic trust? Have you worked on it?

- I was lucky: I never lost it. The feeling of a shoulder and a covered back from the very beginning of the relationship was fundamental for me. And that's what helped me get through different stages, including segments on which we moved at different speeds. I know that my man will never go to deep, thoughtful meanness, that he will act in accordance with his basic principles and his nature. So I perceive any problems and misunderstandings as problems and misunderstandings. If the base is trust and the absence of a knife in the back, then everything else is solvable. I can probably say that my trust is a choice. And I do it every day.

- What about jealousy?

- If in the depths of your soul you understand that anything can happen in life, and you are ready to let go of your man in a situation where his happiness is somewhere else, then the reason for jealousy disappears.

In this regard, the question of lying in a relationship arises. The more you strive to control every step of your partner, the more you dream of merging into a single whole and do not leave him personal space, the more he needs to lie and dodge. Sometimes - so as not to disturb you, sometimes - because it's easier, it happens, because you don't understand how to. I know from my childhood. I grew up with an exclusively controlling mother, where the forces were unequal, and I am not one of those who follow the lead. So, if possible, save your loved one from the very need to lie, give him space, the opportunity not to answer all the questions you ask and not to report on every step. The more you believe in your man and in your man, the better and more comfortable for both of you.

It is very important to learn to respect the decisions of your man. We do not always understand the logic, causes and expected consequences, but not everything needs to be understood with the mind. This is also a necessary component of trust, and this had to be learned.

- Olga, do you and your husband look alike? What conclusion do you draw after so many years together?

No, we are not the same.

So what about being with someone who doesn't look like you? What to do with this dissimilarity?

We are not the same, but we complement each other. I am very interested in his perspective on problems and situations. I'm just interested and warm with him. He constantly generates ideas. It makes you look at many things from a different angle and from the other side. You begin to understand that there can be different answers to the same question, and they both have the right to exist. We can accept that we disagree on some issue. This approach makes life together very interesting and deprives of reasons for conflict.

This dissimilarity can be enjoyed. Get high. Definitely do not try to avoid it or smooth it out (tested - does not work). As with everything, the first step is to recognize where you are different. Does it complement and enrich your shared “we” or are they fundamental differences that it is impossible to be together with? If the differences are fundamental and you are incompatible, the answer is clear - the sooner the couple understands this, the better.

If these are just two different "I", then what is not a task for personal growth? Learn to enjoy your differences, learn to be flexible, learn to be tolerant of your closest person. Probably, next to the dissimilar, you can learn much more. See and know yourself from a completely different perspective.

You started a relationship at a very early age. And these are colossal personal changes - what you are at 18, at 28 or at 48. Absolutely different people, usually. How to continue to love each other despite all these changes?

- While you both grow, change, study, talk about problems, overcome them together, raise children, do joint work, read and discuss, relax, you develop a huge joint history, gratitude to each other for the outstretched hand in time, for warmth, for a hint, for love, for faith… I think that this joint growth only brings them closer. The main thing is that you talk to each other when something goes wrong and don't move in principle opposite sides.

- I was preparing for the meeting and with horror I stumbled upon the thought of my early youth that divorces are normal. Like, if something goes wrong - a divorce. This is fine. I don't know what it was. Or the consequences of the era when a new level of openness and accessibility created this trend. Or lack good examples before my eyes… But I can remember myself at the age of 20, thinking seriously about this. And it seems to be really normal - to disperse, if it really happened. But something else horrified me - along with thoughts about divorces, there was not a single thought that, in fact, building relationships is much more normal. Work on them, strengthening, conscious contribution, the need to go through difficult sections. Have you instilled the idea of ​​such work in your children? And how important is it to talk about it?

“I think it's vital. It is important to teach children this, and even better - to show own example. That is, it is not enough to speak, it is necessary to live your life the way you speak. Children feel falseness a mile away, and absorb emotions and family atmosphere like sponges. What was agony and search for Nikolai and me becomes obvious things for them.

My children and I talked and talk about this a lot, especially in adolescence and now, when they are building their relationships and raising their children. By the way, both say that at some point our example caused difficulties, because the bar was set too high. What is obvious and understandable to them is not obvious to their other halves.

It would be great if moms and society would say things like this more often:

• Happy harmonious relationships do not "happen" - they are built by two loving people.
• Before entering into a long-term relationship, decide on your expectations. Try to understand what is important to you now and in later life (children - their absence, career - home, life in big city- on an island in the ocean, gentle - grasping). It is clear that this will all change many times, but trying to understand your life priorities helps a lot.
• Check the coordinates with your chosen one. Do you agree on the most important issues?
• Your half is a living person, not an ideal. With all the ensuing consequences. In certain situations, you may not like him, and this is normal and does not mean the death of a relationship. It's like with children. I love my children very much, but this does not mean that I like them always and in everything. (Can I explain clearly?)
• He cannot always want what you want (and vice versa).
• Your half is not a copy of you, but a different person. Your task is to hear and understand it. Even though it's probably impossible to fully understand. So accept this difference as a fact of life and don't try to remake (fundamental personality traits, I'm not talking about socks in the hallway).
• The state of happiness and harmony in relationships is not permanent. It comes and goes, but always returns if the couple does not scatter at the first problem situation. And with each such return, feelings become deeper and more tender (we have been through so much together, we have already understood so much about each other).

- Before the first quarrel, it seems that the relationship will always be smooth, small roughness does not count, after the first quarrel it seems that it will never resolve and that this scar is forever. Both you and your partner. Comment from your experience.

- To quarrel without insulting is also a science, it will come with time, but there will also be breakdowns. We perceive the same words differently. One and the same thought can be presented in such a way as to seek a joint solution, or it can be done in such a way that both will lick the scars. The tone is important, the moment is important, how the phrase is built is important. You need to understand why the quarrel happened - because you are tired, sick, overheated, or is there a structural problem in the family that needs to be addressed? It is very important not to get personal. We women often suffer from this.

What can we do about it? How to avoid such passions in the future? How can we talk about the sick person without offending or blaming? Why did you (me) have such a reaction to the remark (question)? I didn’t put such a meaning into it, I didn’t mean it. There can be anything - children's fears, the former negative experience, wrong guesses and thinking out thoughts, our tone and construction of the question. This needs to be talked about. Often not immediately, but when the fuse has cooled down and both of you have calmed down. But it is dangerous to leave such things unthinking.

On the other hand, it is desirable to learn to treat everything easier. (Oh, how long it took me.) Don't try to be perfect, don't try to build perfect relationships, give yourself and the other the right to make mistakes. To understand that swearing and putting up is normal (the question is how this happens), that there will never be complete mutual understanding (this is a myth). Learn not to make an elephant out of a fly. Many "problems" do not need to be corrected or deeply reflected on them, it is better to just forget (as they say, "we drove through, and that's it").

In short, for all the seriousness of the issue, try not to take life together and relationships too seriously. And you don’t need to persistently and endlessly improve everything (yourself, him, relationships), often our imperfections are the highlight that keeps us together.

Woman: "Spare your loved ones from your claims and expectations."

Man: “Don't forget that your husband is a human too. Don't blow his brains out unless absolutely necessary."

Somehow like this.

For a snack, I want to voice an idea that is important for me, which does not directly relate to your questions and, perhaps, will not cause a resonance yet.

Someday in real life we all face death, come to the edge and realize (not with the mind, but with the heart) that we are all here temporarily. Both ourselves and the people we love. After such an “enlightenment” (if you don’t hide your head in the sand from fear) comes a more careful attitude towards yourself and to those who are nearby, and the ability to appreciate the banal little things in life, and most importantly, to receive joy and pleasure from them. It makes life beautiful and filled with love. Maybe if you filter your reactions, relationships, problems, fears through the filter of mortality, then many questions that seem serious will go away by themselves.

Hug tightly.

In addition to the topics, Olga prepared for an independent analysis in the field of relationships and a better understanding of both herself and her man.

Olesya Vlasova

P.S. Friends, for 5 years we have been organizing retreats, expeditions and mountain treks in different parts of Asia. The purpose of our programs is to release the mind and body from tension, restore strength and launch the rhythm of conscious change for the better. Our tools are yoga, meditation, freediving, the practice of silence, the right atmosphere for a complete switch and the good company of like-minded people. If you were looking for a place where you can fully switch and rethink the current “settings” in a qualitative way, we are there.

In the previous tasks for movement in one direction, the movement of bodies began simultaneously from the same point. Consider solving problems for movement in one direction, when the movement of bodies begins at the same time, but from different points.

Let a cyclist and a pedestrian depart from points A and B, the distance between which is 21 km, and go in the same direction: a pedestrian at a speed of 5 km per hour, a cyclist at 12 km per hour

12 km per hour 5 km per hour

A B

The distance between a cyclist and a pedestrian at the start of their movement is 21 km. For an hour of their joint movement in one direction, the distance between them will decrease by 12-5=7 (km). 7 km per hour - the speed of convergence of a cyclist and a pedestrian:

A B

Knowing the speed of approach of a cyclist and a pedestrian, it is easy to find out how many kilometers the distance between them will decrease after 2 hours, 3 hours of their movement in one direction.

7*2=14 (km) - the distance between the cyclist and the pedestrian will decrease by 14 km after 2 hours;

7*3=21 (km) - the distance between the cyclist and the pedestrian will decrease by 21 km after 3 hours.

Every hour the distance between the cyclist and the pedestrian decreases. After 3 hours, the distance between them becomes equal to 21-21=0, i.e. the cyclist overtakes the pedestrian:

A B

In tasks “to catch up” we deal with quantities:

1) the distance between the points from which the simultaneous movement begins;

2) approach speed

3) the time from the moment the movement begins to the moment when one of the moving bodies overtakes the other.

Knowing the value of two of these three quantities, you can find the value of the third quantity.

The table contains the conditions and solutions to problems that can be compiled to “catch up” with a pedestrian cyclist:

We express the relationship between these quantities by the formula. Denote by the distance between the points and, - the speed of approach, the time from the moment of exit to the moment when one body catches up with another.

In catch-up problems, the convergence rate is most often not given, but it can be easily found from the problem data.

Task. A cyclist and a pedestrian left simultaneously in the same direction from two collective farms, the distance between which is 24 km. A cyclist was traveling at a speed of 11 km per hour, and a pedestrian was walking at a speed of 5 km per hour. In how many hours after his exit will the cyclist overtake the pedestrian?

To find how long after his exit the cyclist will catch up with the pedestrian, you need to divide the distance that was between them at the beginning of the movement by the speed of approach; the speed of approach is equal to the difference between the speeds of the cyclist and the pedestrian.

Solution formula: =24: (11-5);=4.

Answer. In 4 hours the cyclist will overtake the pedestrian. Conditions and solutions of inverse problems are written in the table:

Each of these tasks can be solved in other ways, but they will be irrational compared to these solutions.

So let's say our bodies move in the same direction. How many cases do you think there might be for such a condition? That's right, two.

Why is it so? I am sure that after all the examples you will easily figure out how to derive these formulas.

Got it? Well done! It's time to solve the problem.

The fourth task

Kolya goes to work by car at a speed of km/h. Colleague Kolya Vova travels at a speed of km/h. Kolya lives at a distance of km from Vova.

How long will it take Vova to overtake Kolya if they left the house at the same time?

Did you count? Let's compare the answers - it turned out that Vova will catch up with Kolya in hours or minutes.

Let's compare our solutions...

The drawing looks like this:

Similar to yours? Well done!

Since the problem asks how long the guys met and left at the same time, the time they traveled will be the same, as well as the meeting place (in the figure it is indicated by a dot). Making equations, take the time for.

So, Vova made his way to the meeting place. Kolya made his way to the meeting place. It's clear. Now we deal with the axis of movement.

Let's start with the path that Kolya did. Its path () is shown as a segment in the figure. And what does Vova's path () consist of? That's right, from the sum of the segments and, where is the initial distance between the guys, and is equal to the path that Kolya did.

Based on these conclusions, we obtain the equation:

Got it? If not, just read this equation again and look at the points marked on the axis. Drawing helps, doesn't it?

hours or minutes minutes.

I hope that in this example you understand how important the role of well crafted drawing!

And we are smoothly moving on, or rather, we have already moved on to the next step in our algorithm - bringing all quantities to the same dimension.

The rule of three "P" - dimension, reasonableness, calculation.

Dimension.

Not always in tasks the same dimension is given for each participant in the movement (as it was in our easy tasks).

For example, you can meet tasks where it is said that the bodies moved a certain number of minutes, and the speed of their movement is indicated in km / h.

We can't just take and substitute the values ​​in the formula - the answer will be wrong. Even in terms of units of measurement, our answer “will not pass” the test for reasonableness. Compare:

See? With proper multiplication, we also reduce the units of measurement, and, accordingly, we get a reasonable and correct result.

And what happens if we do not translate into one system of measurement? The answer has a strange dimension and % is an incorrect result.

So, just in case, let me remind you the meanings of the basic units of measurement of length and time.

Length units:

centimeter = millimeters

decimeter = centimeters = millimeters

meter = decimeters = centimeters = millimeters

kilometer = meters

Time units:

minute = seconds

hour = minutes = seconds

days = hours = minutes = seconds

Advice: When converting units of measurement related to time (minutes to hours, hours to seconds, etc.), imagine a clock face in your head. It can be seen with the naked eye that minutes is a quarter of the dial, i.e. hours, minutes is a third of the dial, i.e. hours, and a minute is an hour.

And now a very simple task:

Masha rode her bicycle from home to the village at a speed of km/h for minutes. What is the distance between the car house and the village?

Did you count? The correct answer is km.

minutes is an hour, and another minute from an hour (mentally imagined a clock face, and said that minutes is a quarter of an hour), respectively - min \u003d h.

Intelligence.

Do you understand that the speed of a car cannot be km/h, unless, of course, we are talking about a sports car? And even more so, it cannot be negative, right? So, reasonableness, that's about it)

Calculation.

See if your solution "passes" the dimension and reasonableness, and only then check the calculations. It is logical - if there is an inconsistency with dimension and reasonableness, then it is easier to cross out everything and start looking for logical and mathematical errors.

"Love for tables" or "when drawing is not enough"

Far from always, the tasks for movement are as simple as we solved before. Very often, in order to correctly solve a problem, you need to not just draw a competent drawing, but also make a table with all the conditions given to us.

First task

From point to point, the distance between which is km, a cyclist and a motorcyclist left at the same time. It is known that a motorcyclist travels more miles per hour than a cyclist.

Determine the speed of the cyclist if it is known that he arrived at the point a minute later than the motorcyclist.

Here is such a task. Pull yourself together and read it several times. Read? Start drawing - straight line, point, point, two arrows ...

In general, draw, and now let's compare what you got.

Kind of empty, right? We draw a table.

As you remember, all movement tasks consist of components: speed, time and path. It is from these graphs that any table in such problems will consist.

True, we will add one more column - Name about whom we write information - a motorcyclist and a cyclist.

Also indicate in the header dimension, in which you will enter the values ​​\u200b\u200bin there. You remember how important this is, right?

Do you have a table like this?

Now let's analyze everything that we have, and in parallel enter the data into a table and into a figure.

The first thing we have is the path that the cyclist and motorcyclist have traveled. It is the same and equal to km. We bring in!

Let us take the speed of the cyclist as, then the speed of the motorcyclist will be ...

If the solution of the problem does not work with such a variable, it's okay, we'll take another one until we reach the victorious one. This happens, the main thing is not to be nervous!

The table has changed. We have left not filled only one column - time. How to find the time when there is a path and speed?

That's right, divide the path by the speed. Enter it in the table.

So our table has been filled, now you can enter data into the figure.

What can we reflect on it?

Well done. The speed of movement of a motorcyclist and a cyclist.

Let's read the problem again, look at the figure and the completed table.

What data is not shown in the table or in the figure?

Right. The time by which the motorcyclist arrived earlier than the cyclist. We know that the time difference is minutes.

What should we do next? That's right, translate the time given to us from minutes to hours, because the speed is given to us in km / h.

The magic of formulas: writing and solving equations - manipulations that lead to the only correct answer.

So, as you already guessed, now we will make up the equation.

Compilation of the equation:

Look at your table, at the last condition that was not included in it, and think about the relationship between what and what can we put into the equation?

Right. We can make an equation based on the time difference!

Is it logical? The cyclist rode more, if we subtract the time of the motorcyclist from his time, we will just get the difference given to us.

This equation is rational. If you don't know what it is, read the topic "".

We bring the terms to a common denominator:

Let's open the brackets and give like terms: Phew! Got it? Try your hand at the next task.

Equation solution:

From this equation we get the following:

Let's open the brackets and move everything to the left side of the equation:

Voila! We have a simple quadratic equation. We decide!

We received two responses. Look what we got for? That's right, the speed of the cyclist.

We recall the rule "3P", more specifically "reasonableness". Do you understand what I mean? Exactly! Speed ​​cannot be negative, so our answer is km/h.

Second task

Two cyclists set out on a 1-kilometer run at the same time. The first one was driving at a speed that was 1 km/h faster than the second one, and arrived at the finish line hours earlier than the second one. Find the speed of the cyclist who came to the finish line second. Give your answer in km/h.

I recall the solution algorithm:

• Read the problem a couple of times - learn all the details. Got it?
• Start drawing the drawing - in which direction are they moving? how far did they travel? Did you draw?
• Check if all the quantities you have are of the same dimension and start writing out the condition of the problem briefly, making up a table (do you remember what columns are there?).
• While writing all this, think about what to take for? Chose? Record in the table! Well, now it’s simple: we make an equation and solve it. Yes, and finally - remember the "3P"!
• I've done everything? Well done! It turned out that the speed of the cyclist is km / h.

-"What color is your car?" - "She's beautiful!" Correct answers to the questions

Let's continue our conversation. So what is the speed of the first cyclist? km/h? I really hope you're not nodding in the affirmative right now!

Read the question carefully: "What is the speed of first cyclist?

Got what I mean?

Exactly! Received is not always the answer to the question!

Read the questions carefully - perhaps, after finding it, you will need to perform some more manipulations, for example, add km / h, as in our task.

Another point - often in tasks everything is indicated in hours, and the answer is asked to be expressed in minutes, or all the data is given in km, and the answer is asked to be written in meters.

Look at the dimension not only during the solution itself, but also when writing down the answers.

Tasks for movement in a circle

The bodies in the tasks may not necessarily move in a straight line, but also in a circle, for example, cyclists can ride along a circular track. Let's take a look at this problem.

Task #1

A cyclist left the point of the circular track. In minutes he had not yet returned to the checkpoint, and a motorcyclist followed him from the checkpoint. Minutes after departure, he caught up with the cyclist for the first time, and minutes after that he caught up with him for the second time.

Find the speed of the cyclist if the length of the track is km. Give your answer in km/h.

Solution of problem No. 1

Try to draw a picture for this problem and fill in the table for it. Here's what happened to me:

Between meetings, the cyclist traveled the distance, and the motorcyclist -.

But at the same time, the motorcyclist drove exactly one lap more, this can be seen from the figure:

I hope you understand that they didn't actually go in a spiral - the spiral just schematically shows that they go in a circle, passing the same points of the track several times.

Got it? Try to solve the following problems yourself:

Tasks for independent work:

1. Two mo-to-tsik-li-hundreds start-to-tu-yut one-but-time-men-but in one-right-le-ni from two dia-met-ral-but pro-ty-in-po- false points of a circular route, the length of a swarm is equal to km. After how many minutes, mo-the-cycle-lists are equal for the first time, if the speed of one of them is by km / h more than the speed of the other th?
2. From one point of the circle-howl of the highway, the length of some swarm is equal to km, at the same time, in one right-le-ni, there are two motorcyclists. The speed of the first motorcycle is km / h, and minutes after the start, he was ahead of the second motorcycle by one lap. Find the speed of the second motorcycle. Give your answer in km/h.

Solving problems for independent work:

1. Let km / h be the speed of the first mo-to-cycle-li-hundred, then the speed of the second mo-to-cycle-li-hundred is km / h. Let the first time mo-the-cycle-lists be equal in hours. In order for mo-the-cycle-li-stas to be equal, the faster one must overcome them from the beginning distance, equal in lo-vi-not to the length of the route.

   We get that the time is equal to hours = minutes.

2. Let the speed of the second motorcycle be km/h. In an hour, the first motorcycle traveled a kilometer more than the second swarm, respectively, we get the equation:

   The speed of the second motorcyclist is km/h.

Tasks for the course

Now that you're good at solving problems "on land", let's move on to the water and look at the scary problems associated with the current.

Imagine that you have a raft and you lower it into a lake. What is happening to him? Right. It stands because a lake, a pond, a puddle, after all, is stagnant water.

The current velocity in the lake is .

The raft will only move if you start rowing yourself. The speed he gains will be own speed of the raft. No matter where you swim - left, right, the raft will move at the same speed with which you row. It's clear? It's logical.

Now imagine that you are lowering the raft onto the river, turn away to take the rope ..., turn around, and he ... floated away ...

This happens because the river has a flow rate, which carries your raft in the direction of the current.

At the same time, its speed is equal to zero (you are standing in shock on the shore and not rowing) - it moves with the speed of the current.

Got it?

Then answer this question - "How fast will the raft float on the river if you sit and row?" Thinking?

Two options are possible here.

Option 1 - you go with the flow.

And then you swim at your own speed + the speed of the current. The current seems to help you move forward.

2nd option - t You are swimming against the current.

Hard? That's right, because the current is trying to "throw" you back. You make more and more efforts to swim at least meters, respectively, the speed with which you move is equal to your own speed - the speed of the current.

Let's say you need to swim a mile. When will you cover this distance faster? When will you move with the flow or against?

Let's solve the problem and check.

Let's add to our path data on the speed of the current - km/h and on the own speed of the raft - km/h. How much time will you spend moving with and against the current?

Of course, you easily coped with this task! Downstream - an hour, and against the current as much as an hour!

This is the whole essence of the tasks on flow with the flow.

Let's complicate the task a little.

Task #1

A boat with a motor sailed from point to point in an hour, and back in an hour.

Find the speed of the current if the speed of the boat in still water is km/h

Solution of problem No. 1

Let's denote the distance between the points as, and the speed of the current as.

We see that the boat makes the same path, respectively:

What did we charge for?

Flow speed. Then this will be the answer :)

The speed of the current is km/h.

Task #2

The kayak went from point to point, located km away. After staying at point for an hour, the kayak set off and returned to point c.

Determine (in km/h) the own speed of the kayak if it is known that the speed of the river is km/h.

Solution of problem No. 2

So let's get started. Read the problem several times and draw a picture. I think you can easily solve this on your own.

Are all quantities expressed in the same form? No. The rest time is indicated both in hours and in minutes.

Converting this to hours:

hour minutes = h.

Now all quantities are expressed in one form. Let's start filling out the table and looking for what we'll take for.

Let be the own speed of the kayak. Then, the speed of the kayak downstream is equal, and against the current is equal.

Let's write this data, as well as the path (as you understand, it is the same) and the time expressed in terms of path and speed, in a table:

Let's calculate how much time the kayak spent on its trip:

Did she swim all hours? Rereading the task.

No, not all. She had a rest of an hour of minutes, respectively, from the hours we subtract the rest time, which we have already translated into hours:

h kayak really floated.

Let's bring all the terms to a common denominator:

We open the brackets and give like terms. Next, we solve the resulting quadratic equation.

With this, I think you can also handle it on your own. What answer did you get? I have km/h.

Summing up

ADVANCED LEVEL

Movement tasks. Examples

Consider examples with solutionsfor each type of task.

moving with the flow

One of the most simple tasks - tasks for the movement on the river. Their whole essence is as follows:

• if we move with the flow, the speed of the current is added to our speed;
• if we move against the current, the speed of the current is subtracted from our speed.

Example #1:

The boat sailed from point A to point B in hours and back in hours. Find the speed of the current if the speed of the boat in still water is km/h.

Solution #1:

Let's denote the distance between the points as AB, and the speed of the current as.

We will enter all the data from the condition in the table:

For each row of this table, you need to write the formula:

In fact, you don't have to write equations for each of the rows in the table. We see that the distance traveled by the boat back and forth is the same.

So we can equate the distance. To do this, we immediately use distance formula:

Often it is necessary to use formula for time:

Example #2:

A boat travels a distance in km against the current for an hour longer than with the current. Find the speed of the boat in still water if the speed of the current is km/h.

Solution #2:

Let's try to write an equation. The time upstream is one hour longer than the time downstream.

It is written like this:

Now, instead of each time, we substitute the formula:

We got the usual rational equation, we solve it:

Obviously the speed cannot be negative number so the answer is km/h.

Relative motion

If some bodies are moving relative to each other, it is often useful to calculate their relative speed. It is equal to:

• the sum of velocities if the bodies move towards each other;
• speed difference if the bodies are moving in the same direction.

Example #1

From points A and B, two cars left simultaneously towards each other with speeds of km/h and km/h. In how many minutes will they meet? If the distance between points is km?

I solution way:

Relative speed of cars km/h. This means that if we are sitting in the first car, it seems to be stationary, but the second car is approaching us at a speed of km/h. Since the distance between cars is initially km, the time after which the second car will pass the first:

Solution 2:

The time from the start of the movement to the meeting at the cars is obviously the same. Let's designate it. Then the first car drove the way, and the second -.

In total, they traveled all km. Means,

Other motion tasks

Example #1:

A car left point A for point B. Simultaneously with it, another car left, which traveled exactly half the way at a speed of km/h less than the first one, and the second half of the way it drove at a speed of km/h.

As a result, the cars arrived at point B at the same time.

Find the speed of the first car if it is known to be greater than km/h.

Solution #1:

To the left of the equal sign, we write the time of the first car, and to the right - the second:

Simplify the expression on the right side:

We divide each term by AB:

It turned out the usual rational equation. Solving it, we get two roots:

Of these, only one is larger.

Answer: km/h.

Example #2

A cyclist left point A of the circular track. After a few minutes, he had not yet returned to point A, and a motorcyclist followed him from point A. Minutes after departure, he caught up with the cyclist for the first time, and minutes after that he caught up with him for the second time. Find the speed of the cyclist if the length of the track is km. Give your answer in km/h.

Solution:

Here we will equate the distance.

Let the speed of the cyclist be, and the speed of the motorcyclist -. Until the moment of the first meeting, the cyclist was on the road for minutes, and the motorcyclist -.

In doing so, they traveled equal distances:

Between meetings, the cyclist traveled the distance, and the motorcyclist -. But at the same time, the motorcyclist drove exactly one lap more, this can be seen from the figure:

I hope you understand that they didn't actually go in a spiral - the spiral just schematically shows that they go in a circle, passing the same points of the track several times.

We solve the resulting equations in the system:

SUMMARY AND BASIC FORMULA

1. Basic formula

2. Relative motion

• This is the sum of the speeds if the bodies are moving towards each other;
• speed difference if the bodies are moving in the same direction.

3. Move with the flow:

• If we move with the current, the speed of the current is added to our speed;
• if we move against the current, the speed of the current is subtracted from the speed.

We have helped you deal with the tasks of movement...

Now it's your turn...

If you carefully read the text and solved all the examples yourself, we are ready to argue that you understood everything.

And this is already half way.

Write below in the comments if you figured out the tasks for movement?

Which cause the greatest difficulty?

Do you understand that tasks for "work" are almost the same thing?

Write to us and good luck on your exams!