This is a vector physical quantity, numerically equal to the limit to which the average speed tends over an infinitely small period of time:
In other words, the instantaneous speed is the radius vector in time.
The instantaneous velocity vector is always directed tangentially to the body trajectory in the direction of body movement.
Instant Speed gives accurate information about the movement at a certain point in time. For example, while driving in a car at some point in time, the driver looks at the speedometer and sees that the device shows 100 km / h. After a while, the speedometer needle points to 90 km / h, and after a few minutes - to 110 km / h. All the listed speedometer readings are the values of the instantaneous speed of the car at certain points in time. The speed at each moment of time and at each point of the trajectory must be known when docking space stations, when landing aircraft, etc.
Does the concept of "instantaneous speed" physical meaning? Speed is a characteristic of change in space. However, in order to determine how the movement has changed, it is necessary to observe the movement for some time. Even the most advanced instruments for measuring speed, such as radar installations, measure speed over a period of time - albeit a fairly small one, but this is still a finite time interval, and not a moment in time. The expression "the speed of a body in this moment time" from the point of view of physics is not correct. However, the concept of instantaneous speed is very convenient in mathematical calculations, and it is constantly used.
Examples of solving problems on the topic "Instant speed"
EXAMPLE 1
EXAMPLE 2
| Exercise | The law of motion of a point along a straight line is given by the equation. Find the instantaneous speed of the point 10 seconds after the start of movement. |
| Solution | The instantaneous velocity of a point is the radius vector in time. Therefore, for the instantaneous speed, we can write: 10 seconds after the start of movement, the instantaneous speed will have the value: |
| Answer | 10 seconds after the start of movement, the instantaneous speed of the point is m/s. |
EXAMPLE 3
| Exercise | The body moves in a straight line so that its coordinate (in meters) changes according to the law. In how many seconds after the start of motion will the body stop? |
| Solution | Find the instantaneous speed of the body: |
If material point is in motion, its coordinates are subject to change. This process can be fast or slow.
Definition 1
The value that characterizes the rate of change in the position of the coordinate is called speed.
Definition 2
average speed is a vector quantity, numerically equal to the displacement per unit time, and co-directional with the displacement vector υ = ∆ r ∆ t ; υ ∆ r .
Picture 1 . The average speed is co-directed to the movement
The modulus of the average speed along the path is equal to υ = S ∆ t .
Instantaneous speed characterizes the movement at a certain point in time. The expression "velocity of a body at a given time" is considered incorrect, but applicable in mathematical calculations.
Definition 3
The instantaneous speed is the limit to which the average speed υ tends when the time interval ∆t tends to 0:
υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ .
The direction of the vector υ is tangent to the curvilinear trajectory, because the infinitesimal displacement d r coincides with the infinitesimal element of the trajectory d s .
Figure 2. Instantaneous velocity vector υ
The existing expression υ = l i m ∆ t ∆ r ∆ t = d r d t = r ˙ in Cartesian coordinates is identical to the equations proposed below:
υ x = d x d t = x ˙ υ y = d y d t = y ˙ υ z = d z d t = z ˙ .
The record of the modulus of the vector υ will take the form:
υ \u003d υ \u003d υ x 2 + υ y 2 + υ z 2 \u003d x 2 + y 2 + z 2.
To go from Cartesian rectangular coordinates to curvilinear, apply the rules of differentiation of complex functions. If the radius vector r is a function of curvilinear coordinates r = r q 1 , q 2 , q 3 , then the velocity value is written as:
υ = d r d t = ∑ i = 1 3 ∂ r ∂ q i ∂ q i ∂ r = ∑ i = 1 3 ∂ r ∂ q i q ˙ i .
Figure 3. Displacement and instantaneous velocity in curvilinear coordinate systems
For spherical coordinates, suppose that q 1 = r ; q 2 \u003d φ; q 3 \u003d θ, then we get υ presented in this form:
υ = υ r e r + υ φ e φ + υ θ φ θ , where υ r = r ˙ ; υ φ = r φ ˙ sin θ ; υ θ = r θ ˙ ; r ˙ = d r d t ; φ ˙ = d φ d t ; θ ˙ = d θ d t ; υ \u003d r 1 + φ 2 sin 2 θ + θ 2.
Definition 4
instantaneous speed call the value of the derivative of the function of movement in time at a given moment, associated with the elementary movement by the relation d r = υ (t) d t
Example 1
Given the law rectilinear motion points x (t) = 0 , 15 t 2 - 2 t + 8 . Determine its instantaneous speed 10 seconds after the start of movement.
Solution
The instantaneous velocity is usually called the first derivative of the radius vector with respect to time. Then its entry will look like:
υ (t) = x ˙ (t) = 0 . 3 t - 2 ; υ (10) = 0 . 3 × 10 - 2 = 1 m/s.
Answer: 1 m/s.
Example 2
The movement of a material point is given by the equation x = 4 t - 0 , 05 t 2 . Calculate the moment of time t about with t when the point stops moving, and its average ground speed υ.
Solution
Calculate the equation of instantaneous speed, substitute numerical expressions:
υ (t) = x ˙ (t) = 4 - 0 , 1 t .
4 - 0 , 1 t = 0 ; t about with t \u003d 40 s; υ 0 = υ (0) = 4; υ = ∆ υ ∆ t = 0 - 4 40 - 0 = 0 , 1 m / s.
Answer: the set point will stop after 40 seconds; the value of the average speed is 0.1 m/s.
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And why is it needed. We already know what a frame of reference, relativity of motion and a material point are. Well, it's time to move on! Here we will look at the basic concepts of kinematics, put together the most useful formulas on the basics of kinematics, and present practical example problem solving.
Let's solve the following problem: A point moves in a circle with a radius of 4 meters. The law of its motion is expressed by the equation S=A+Bt^2. A=8m, B=-2m/s^2. At what point in time is the normal acceleration of a point equal to 9 m/s^2? Find the speed, tangential and total acceleration of the point for this moment in time.
Solution: we know that in order to find the speed, we need to take the first time derivative of the law of motion, and the normal acceleration is equal to the private square of the speed and the radius of the circle along which the point moves. Armed with this knowledge, we find the desired values.
Need help solving problems? A professional student service is ready to provide it.
Methods for specifying the movement of a point.
Set Point Movement - this means to indicate the rule by which at any time you can determine its position in a given reference frame.
The mathematical expression for this rule is called the law of motion , or motion equation points.
There are three ways to specify the movement of a point:
vector;
coordinate;
natural.
To set the movement in a vector way, need to:
à select a fixed center;
à determine the position of the point using the radius vector , starting at the fixed center and ending at the moving point M;
à define this radius vector as a function of time t: 
.
Expression
called vector law of motion dots, or vector equation of motion.
!! Radius vector - this is the distance (vector modulus) + direction from the center O to the point M, which can be determined in different ways, for example, by angles with given directions.
To set movement coordinate way , need to:
à select and fix a coordinate system (any: Cartesian, polar, spherical, cylindrical, etc.);
à determine the position of the point using the appropriate coordinates;
à set these coordinates as functions of time t.
In the Cartesian coordinate system, therefore, it is necessary to specify the functions
In the polar coordinate system, the polar radius and polar angle should be defined as functions of time:
In general, with the coordinate method of setting, one should set as a function of time those coordinates with which the current position of the point is determined.
To be able to set the movement of the point natural way, you need to know it trajectory . Let us write down the definition of the trajectory of a point.
trajectory point is called set of its positions for any period of time(usually from 0 to +¥).
In the example with the wheel rolling on the road, the trajectory of point 1 is cycloid, and points 2 – roulette; in the reference frame associated with the center of the wheel, the trajectories of both points are circles.
To set the movement of a point in a natural way, you need to:
à know the trajectory of the point;
à on the trajectory, select the origin and the positive direction;
à determine the current position of the point by the length of the trajectory arc from the origin to this current position;
à specify this length as a function of time.
An expression that defines the above function,
called the law of motion of a point along a trajectory, or natural equation of motion points.
Depending on the type of function (4), a point along the trajectory can move in different ways.
3. Point trajectory and its definition.
The definition of the concept of "point trajectory" was given earlier in question 2. Let's consider the question of determining the trajectory of a point with different ways of specifying motion.
natural way: the trajectory must be given, so it is not necessary to find it.
Vector way: you need to switch to the coordinate method according to the equalities
Coordinate method : it is necessary to exclude the time t from the equations of motion (2), or (3).
The coordinate equations of motion define the trajectory parametrically, through the parameter t (time). To obtain an explicit equation for the curve, the parameter must be excluded from the equations.
After excluding time from equations (2), two equations of cylindrical surfaces are obtained, for example, in the form
The intersection of these surfaces will be the trajectory of the point.
When a point moves along a plane, the problem is simplified: after eliminating time from the two equations
the trajectory equation will be in one of the following forms:
When will be, so the trajectory of the point will be the right branch of the parabola:
It follows from the equations of motion that
therefore, the trajectory of the point will be the part of the parabola located in the right half-plane:
Then we get
Since then the entire ellipse will be the trajectory of the point.
At 
the center of the ellipse will be at the origin O; when we get a circle; the parameter k does not affect the shape of the ellipse, it determines the speed of the point moving along the ellipse. If cos and sin are interchanged in the equations, then the trajectory will not change (the same ellipse), but the initial position of the point and the direction of movement will change.
The speed of a point characterizes the “speed” of changing its position. Formally: speed - movement of a point per unit of time.
Precise definition.
Then 
Attitude
