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

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 called curvilinear. If the trajectory is flat, then the motion of the point is called flat.

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

c) the direction of the positive count of 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 periods 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 periods 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). Let's consider point A, which moves along a given trajectory according to a certain law s = s(t) (Figure 12):

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

The true speed of a 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

Dimension of point speed: (v) = length/time, for example, m/s. If the point moves in the direction of increasing curvilinear coordinate s, then ds > 0, and therefore v > 0, otherwise ds< 0 и v < 0.

Point acceleration. The change in speed per unit time is determined by acceleration. Let's 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 a speed v, and in position A 1 - a speed v 1 (Figure 13). those. the speed of the point changed in magnitude and direction. We find the geometric difference of speeds Δv by constructing the vector v 1 from point A.

The acceleration of a point is the vector “, which is 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 tangent and normal to the trajectory of motion. Tangential acceleration a 1 coincides in direction with the speed during accelerated motion or is opposite to it during replaced motion. It characterizes the change in speed and is equal to the derivative of the speed with respect to time

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 velocity of the point to the radius of curvature of the trajectory at the point in question.

Normal acceleration characterizes the change in speed along
direction.

Total acceleration value: , m/s 2

Types of point motion depending on acceleration.

Uniform linear movement(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 uneven 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 infinitesimal period of time:

In other words, instantaneous speed is the radius vector over time.

The instantaneous velocity vector is always directed tangentially to the body's trajectory in the direction of the body's movement.

Instantaneous speed provides precise information about movement at a specific point in time. For example, when driving a car at some point in time, the driver looks at the speedometer and sees that the device shows 100 km/h. After some time, the speedometer needle points to 90 km/h, and a few minutes later – to 110 km/h. All of 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 planes, etc.

Does the concept of "instantaneous speed" physical meaning? Velocity 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 quite small, but this is still a finite time interval, and not a moment in time. The expression "velocity 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 is constantly used.

Examples of solving problems on the topic “Instantaneous speed”

EXAMPLE 1

EXAMPLE 2

EXAMPLE 3

The speed of a point is a vector that determines at any given moment in time the speed and direction of movement of the point.

The speed of uniform motion 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-path; t- time.

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

When rectilinear motion the velocity vector is directed along the trajectory in the direction of its movement.

If a point travels unequal paths in equal periods of time, then this movement is called uneven. Speed ​​is a variable quantity and is a function of time.

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

Let's consider point M, which moves along a curvilinear trajectory specified by the law

Over a period of time?t, point M will move to position M1 along the arc MM 1. If the time period?t is small, then arc MM 1 can be replaced by a chord and, to a 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 at?t> 0

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

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

Point acceleration

Note that in the general case, when moving along a curved path, 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 in speed over time. If during the time interval?t the speed changes by an amount, then the average acceleration

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

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

Acceleration dimension

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 tangentially and normal to the trajectory of the point.

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

Let us denote the component limits by and.

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

This acceleration always coincides with the direction of the velocity, 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 a point is directed perpendicular to the tangent to the trajectory at a 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 in the speed of the point over the considered period?t of time, 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 velocity. 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 during uneven curvilinear motion of a point is composed geometrically of the tangential and normal accelerations.