And why is it needed. We already know what a frame of reference is, the relativity of motion and material point. 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.

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The speed of a point is a vector that determines in each this moment time, the speed and direction of movement of the point.

The speed of uniform movement is determined by the ratio of the path traveled by a point in a certain period of time to the value of this period of time.

Speed; S- way; t- time.

The speed is measured in units of length divided by a unit of time: m/s; cm/s; km/h, etc.

In the case of rectilinear motion, the velocity vector is directed along the trajectory in the direction of its motion.

If a point travels unequal paths in equal intervals of time, then this movement is called uneven. Velocity is a variable and is a function of time.

The average speed of a point over a given period of time is the speed of such a uniform rectilinear motion at which the point would receive the same movement during this period of time as in its considered movement.

Consider a point M that moves along a curvilinear trajectory given by the law

During the time interval? t, the point M will move to the position M 1 along the arc MM 1. If the time interval? t is small, then the arc MM 1 can be replaced by a chord and, in the first approximation, find the average speed of the point

This speed is directed along the chord from point M to point M 1 . We find the true speed by going to the limit when? t> 0

When?t> 0, the direction of the chord in the limit coincides with the direction of the tangent to the trajectory at the point M.

Thus, the value of the speed of a point is defined as the limit of the ratio of the path increment to the corresponding time interval as the latter tends to zero. The direction of the velocity coincides with the tangent to the trajectory at the given point.

point acceleration

Note that in the general case, when moving along a curvilinear trajectory, the speed of a point changes both in direction and in magnitude. The change in speed per unit time is determined by acceleration. In other words, the acceleration of a point is a quantity that characterizes the rate of change of speed over time. If for a time interval? t the speed changes by a value, then the average acceleration

The true acceleration of a point at a given time t is the value to which the average acceleration tends when? t\u003e 0, that is

With a time interval tending to zero, the acceleration vector will change both in magnitude and direction, tending to its limit.

Dimension of acceleration

Acceleration can be expressed in m/s 2 ; cm/s 2 etc.

In the general case, when the motion of a point is given in a natural way, the acceleration vector is usually decomposed into two components directed along the tangent and along the normal to the point's trajectory.

Then the acceleration of a point at time t can be represented as

Let us denote the constituent limits by and.

The direction of the vector does not depend on the size of the time interval?t.

This acceleration always coincides with the direction of speed, that is, it is directed tangentially to the trajectory of the point and is therefore called tangential or tangential acceleration.

The second component of the acceleration of the point is directed perpendicular to the tangent to the trajectory at the given point towards the concavity of the curve and affects the change in the direction of the velocity vector. This component of acceleration is called normal acceleration.

Since the numerical value of the vector is equal to the increment of the point velocity over the considered time interval?t, then the numerical value of the tangential acceleration

The numerical value of the tangential acceleration of a point is equal to the time derivative of the numerical value of the speed. The numerical value of the normal acceleration of a point is equal to the square of the point's speed divided by the radius of curvature of the trajectory at the corresponding point on the curve

The total acceleration in case of non-uniform curvilinear motion of a point is geometrically composed of the tangential and normal accelerations.

Mechanical motion is a change over time in the position in space of points and bodies relative to any main body with which the frame of reference is attached. Kinematics studies mechanical movement points and bodies, regardless of the forces that cause these movements. Any movement, like rest, is relative and depends on the choice of the frame of reference.

The trajectory of a point is a continuous line described by a moving point. If the trajectory is a straight line, then the movement of the point is called rectilinear, and if it is a curve, then it is curvilinear. If the trajectory is flat, then the motion of the point is called flat.

The motion of a point or body is considered given or known if for each moment of time (t) it is possible to indicate the position of the point or body relative to the selected coordinate system.

The position of a point in space is determined by the task:

a) point trajectories;

b) the beginning of O 1 distance reading along the trajectory (Figure 11): s = O 1 M - curvilinear coordinate of the point M;

c) the direction of the positive reading of the distances s;

d) equation or law of motion of a point along a trajectory: S = s(t)

Point speed. If a point travels equal distances in equal intervals of time, then its motion is called uniform. The speed of uniform motion is measured by the ratio of the path z traveled by a point in a certain period of time to the value of this period of time: v = s / 1. If a point travels unequal paths in equal intervals of time, then its movement is called uneven. The speed in this case is also variable and is a function of time: v = v(t). Consider point A, which moves along a given trajectory according to a certain law s = s(t) (Figure 12):

For a period of time t t. A moved to position A 1 along the arc AA. If the time interval Δt is small, then the arc AA 1 can be replaced by a chord and, in the first approximation, the average speed of the point movement v cp = Ds/Dt can be found. The average speed is directed along the chord from t. A to t. A 1.

The true speed of the point is directed tangentially to the trajectory, and its algebraic value is determined by the first derivative of the path with respect to time:

v = limΔs/Δt = ds/dt

Unit of point velocity: (v) = length/time, eg m/s. If the point moves in the direction of increasing curvilinear coordinate s, then ds > 0, and hence v > 0, otherwise ds< 0 и v < 0.

Point acceleration. The change in speed per unit time is determined by acceleration. Consider the movement of point A along a curvilinear trajectory in time Δt from position A to position A 1 . In position A, the point had speed v , and in position A 1 - speed v 1 (Figure 13). those. the speed of the dot changed in magnitude and direction. We find the geometric difference, velocities Δv, by constructing a vector v 1 from point A.

The acceleration of a point is called the vector ", equal to the first derivative of the point's velocity vector with respect to time:

The found acceleration vector a can be decomposed into two mutually perpendicular components but the tangent and the normal to the motion trajectory . The tangential acceleration a 1 coincides in direction with the speed during accelerated movement or is opposite to it during the replaced movement. It characterizes the change in the speed value and is equal to the time derivative of the speed value

The normal acceleration vector a is directed along the normal (perpendicular) to the curve towards the concavity of the trajectory, and its modulus is equal to the ratio of the square of the point velocity to the radius of curvature of the trajectory at the point under consideration.

Normal acceleration characterizes the change in speed along
direction.

Full acceleration value: , m/s 2

Types of point movement depending on acceleration.

Uniform rectilinear motion (motion by inertia) is characterized by the fact that the speed of movement is constant, and the radius of curvature of the trajectory is equal to infinity.

That is, r = ¥, v = const, then ; and therefore . So, when a point moves by inertia, its acceleration is zero.

Rectilinear non-uniform movement. The radius of curvature of the trajectory is r = ¥, and n = 0, therefore, a = a t and a = a t = dv/dt.

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 "velocity of a body at a given moment of 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

EXAMPLE 3