Mechanical movement is called the change over time in the position in space of points and bodies relative to any main body to which the frame of reference is attached. Kinematics studies the mechanical movement of points and bodies, regardless of the forces that cause these movements. Any movement, like rest, is relative and depends on the choice of a 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 called curvilinear. If the trajectory is flat, then the movement of the point is called flat.
The motion of a point or body is considered given or known if for each moment in time (t) you can specify 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 counting the distance along the trajectory (Figure 11): s = O 1 M - curvilinear coordinate of point M;
c) directions of positive counting of distances s;
d) equations or law of motion of a point along a trajectory: S = s (t)
Point speed. If a point travels equal path segments in equal time intervals, then its movement is called uniform. The speed of uniform movement is measured by the ratio of the path z traversed by a point in a certain period of time to the value of this period of time: v = s / 1. If a point travels uneven paths over 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 some law s = s (t) (Figure 12):
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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 value of the average speed of movement of the point v cp = Ds / Dt can be found. 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 velocity: (v) = length / time, for example, m / s. If the point moves towards an increase in the curvilinear coordinate s, then ds> 0, and therefore v> 0, and otherwise ds< 0 и v < 0.
Point acceleration. The change in speed per unit of time is determined by the acceleration. Consider the movement of point A along a curved 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 has changed in magnitude and direction. We find the geometric difference, the velocities Δv, by constructing a vector v 1 from the point A.
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The acceleration of a point is called a vector "equal to the first derivative of the velocity vector of a point with respect to time:
The found acceleration vector a can be decomposed into two mutually perpendicular components but the tangent and normal to the trajectory of motion. The tangential acceleration a 1 coincides in direction with the speed during accelerated motion or opposite to it when the motion is replaced. It characterizes the change in the magnitude of the speed and is equal to the derivative of the magnitude of the speed in 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 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 movement(motion by inertia) is characterized by the fact that the speed of motion 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 infinitely small period of time:
In other words, the instantaneous velocity is the radius vector in time.
The instantaneous velocity vector is always directed tangentially to the trajectory of the body in the direction of the movement of the body.
Instantaneous speed provides accurate information about movement at a specific point in time. For example, when 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 a few minutes later - to 110 km / h. All of the listed speedometer readings are the values of the instantaneous vehicle speed 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" have a physical meaning? Velocity is a characteristic of change in space. However, in order to determine how the displacement has changed, it is necessary to observe the movement for some time. Even the most advanced speed measuring devices, such as radar systems, measure speed over a period of time - albeit small enough, but it is still a finite time interval, not a moment in time. The expression “the speed of a body at a given moment of time” is not correct from the point of view of physics. However, the concept of instantaneous speed is very convenient in mathematical calculations, and it is constantly used.
Examples of solving problems on the topic "Instantaneous 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 the movement. |
| Solution | The instantaneous speed of a point is the radius vector in time. Therefore, for the instantaneous speed, you can write: 10 seconds after the start of movement, the instantaneous speed will have a value: |
| Answer | In 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. How many seconds after the start of movement will the body stop? |
| Solution | Let's find the instantaneous speed of the body: |
Methods for specifying the movement of a point.
Set point movement - it means to indicate the rule according to which at any moment of time it is possible to determine its position in a given frame of reference.
The mathematical expression for this rule is called law of motion , or equation of motion points.
There are three ways to define the movement of a point:
vector;
coordinate;
natural.
To set motion in a vector way, need to:
à choose a fixed center;
à the position of the point is determined 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 points, or vector equation of motion.
!! Radius vector Is the distance (modulus of the vector) + direction from the center O to the point M, which can be determined in different ways, for example, angles with given directions.
To set motion 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 corresponding coordinates;
à set these coordinates as functions of time t.
In a Cartesian coordinate system, therefore, you must specify the functions
In a polar coordinate system, the polar radius and polar angle should be defined as functions of time:
In general, in the coordinate method of specifying, the coordinates that determine the current position of the point should be set as a function of time.
So that you can set the movement of the point in a natural way you need to know her trajectory ... Let's write down the definition of the trajectory of the point.
Trajectory points called many of its positions for any period of time(usually between 0 and + ¥).
In the example with a wheel rolling on the road, the trajectory of point 1 is cycloid, and points 2 - roll; in the frame of reference associated with the center of the wheel, the trajectories of both points - circles.
To set the movement of a point in a natural way, you need:
à know the trajectory of the point;
à select the origin and the positive direction on the trajectory;
à determine the current position of a point by the length of the trajectory arc from the origin to this current position;
à indicate this length as a function of time.
An expression defining the above function,
are called the law of motion of a point along a trajectory, or natural equation of motion points.
Depending on the type of function (4), the point along the trajectory can move in different ways.
3. Trajectory of a point and its definition.
The definition of the concept of "trajectory of a point" was given earlier in question 2. Consider the question of determining the trajectory of a point for different ways of specifying the movement.
Natural way: the trajectory must be specified, so you don't need to find it.
Vector way: you need to go to the coordinate method according to the equalities
Coordinate way: 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 excluding time from two equations
the trajectory equation will be obtained in one of the following forms:
When will be, therefore, 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 path of the point.
At 
the center of the ellipse will be at the origin of coordinates O; when we get a circle; the parameter k does not affect the shape of the ellipse; the speed of the point along the ellipse depends on it. 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 the change in its position. Formally: speed - movement of a point per unit of time.
Precise definition.
Then 
Attitude
1.2. Straight motion
1.2.4. average speed
A material point (body) retains its speed unchanged only with uniform rectilinear motion. If the movement is uneven (including equally variable), then the speed of the body changes. This movement is characterized by an average speed. Distinguish between average travel speed and average ground speed.
Average travel speed is a vector physical quantity, which is determined by the formula
v → r = Δ r → Δ t,
where Δ r → is the displacement vector; ∆t is the time interval during which this movement took place.
Average ground speed is a scalar physical quantity and is calculated by the formula
v s = S total t total,
where S total = S 1 + S 1 + ... + S n; t total = t 1 + t 2 + ... + t N.
Here S 1 = v 1 t 1 - the first section of the path; v 1 - the speed of passage of the first section of the path (Fig. 1.18); t 1 - the time of movement on the first section of the path, etc.
Rice. 1.18
Example 7. One quarter of the way the bus moves at a speed of 36 km / h, the second quarter of the way - 54 km / h, the rest of the way - at a speed of 72 km / h. Calculate the average road speed of the bus.
Solution. The total path traveled by the bus is denoted by S:
S total = S.
S 1 = S / 4 - the path traveled by the bus on the first section,
S 2 = S / 4 - the path traveled by the bus on the second section,
S 3 = S / 2 - the path traveled by the bus in the third section.
The bus travel time is determined by the formulas:
- in the first section (S 1 = S / 4) -
t 1 = S 1 v 1 = S 4 v 1;
- in the second section (S 2 = S / 4) -
t 2 = S 2 v 2 = S 4 v 2;
- in the third section (S 3 = S / 2) -
t 3 = S 3 v 3 = S 2 v 3.
The total travel time of the bus is:
t total = t 1 + t 2 + t 3 = S 4 v 1 + S 4 v 2 + S 2 v 3 = S (1 4 v 1 + 1 4 v 2 + 1 2 v 3).
v s = S total t total = S S (1 4 v 1 + 1 4 v 2 + 1 2 v 3) =
1 (1 4 v 1 + 1 4 v 2 + 1 2 v 3) = 4 v 1 v 2 v 3 v 2 v 3 + v 1 v 3 + 2 v 1 v 2.
v s = 4 ⋅ 36 ⋅ 54 ⋅ 72 54 ⋅ 72 + 36 ⋅ 72 + 2 ⋅ 36 ⋅ 54 = 54 km / h.
Example 8. One fifth of the time a city bus spends on stops, the rest of the time it moves at a speed of 36 km / h. Determine the average road speed of the bus.
Solution. The total travel time of the bus on the route is denoted by t:
t total = t.
t 1 = t / 5 - time spent on stops,
t 2 = 4t / 5 - bus travel time.
The route traveled by the bus:
- for time t 1 = t / 5 -
S 1 = v 1 t 1 = 0,
since the speed of the bus v 1 in this time interval is equal to zero (v 1 = 0);
- in time t 2 = 4t / 5 -
S 2 = v 2 t 2 = v 2 4 t 5 = 4 5 v 2 t,
where v 2 is the speed of the bus at a given time interval (v 2 = = 36 km / h).
The total bus route is:
S total = S 1 + S 2 = 0 + 4 5 v 2 t = 4 5 v 2 t.
We calculate the average ground speed of the bus using the formula
v s = S total t total = 4 5 v 2 t t = 4 5 v 2.
The calculation gives the value of the average ground speed:
v s = 4 5 ⋅ 36 = 30 km / h.
Example 9. The equation of motion of a material point has the form x (t) = (9.0 - 6.0t + 2.0t 2) m, where the coordinate is given in meters, time - in seconds. Determine the average ground speed and the value of the average speed of movement of a material point in the first three seconds of movement.
Solution. For determining average travel speed it is necessary to calculate the movement of the material point. The modulus of movement of a material point in the time interval from t 1 = 0 s to t 2 = 3.0 s is calculated as the difference of coordinates:
| Δ r → | = | x (t 2) - x (t 1) | ,
Substituting values into the formula for calculating the displacement modulus gives:
| Δ r → | = | x (t 2) - x (t 1) | = 9.0 - 9.0 = 0 m.
Thus, the displacement of the material point is zero. Therefore, the modulus of the average movement speed is also zero:
| v → r | = | Δ r → | t 2 - t 1 = 0 3.0 - 0 = 0 m / s.
For determining average ground speed it is necessary to calculate the path traversed by the material point during the time interval from t 1 = 0 s to t 2 = 3.0 s. The point's movement is uniformly slow, so you need to find out if the stopping point falls within the specified interval.
To do this, we write down the law of change in the speed of a material point over time in the form:
v x = v 0 x + a x t = - 6.0 + 4.0 t,
where v 0 x = −6.0 m / s is the projection of the initial velocity onto the Ox axis; a x = = 4.0 m / s 2 - projection of acceleration onto the specified axis.
Find a stopping point from the condition
v (τ rest) = 0,
those.
τ rest = v 0 a = 6.0 4.0 = 1.5 s.
The stopping point falls within the time interval from t 1 = 0 s to t 2 = 3.0 s. Thus, the traversed path is calculated by the formula
S = S 1 + S 2,
where S 1 = | x (τ rest) - x (t 1) | - the path traversed by a material point to a stop, i.e. for the time from t 1 = 0 s to τ rest = 1.5 s; S 2 = | x (t 2) - x (τ rest) | - the path traversed by the material point after stopping, i.e. for the time from τ rest = 1.5 s to t 1 = 3.0 s.
Let's calculate the values of the coordinates at the specified times:
x (t 1) = 9.0 - 6.0 t 1 + 2.0 t 1 2 = 9.0 - 6.0 ⋅ 0 + 2.0 ⋅ 0 2 = 9.0 m;
x (τ rest) = 9.0 - 6.0 τ rest + 2.0 τ rest 2 = 9.0 - 6.0 ⋅ 1.5 + 2.0 ⋅ (1.5) 2 = 4.5 m ;
x (t 2) = 9.0 - 6.0 t 2 + 2.0 t 2 2 = 9.0 - 6.0 ⋅ 3.0 + 2.0 ⋅ (3.0) 2 = 9.0 m ...
The coordinate values allow you to calculate the paths S 1 and S 2:
S 1 = | x (τ rest) - x (t 1) | = | 4.5 - 9.0 | = 4.5 m;
S 2 = | x (t 2) - x (τ rest) | = | 9.0 - 4.5 | = 4.5 m,
as well as the total distance traveled:
S = S 1 + S 2 = 4.5 + 4.5 = 9.0 m.
Therefore, the desired value of the average ground speed of a material point is
v s = S t 2 - t 1 = 9.0 3.0 - 0 = 3.0 m / s.
Example 10. The graph of the dependence of the projection of the speed of a material point on time is a straight line and passes through the points (0; 8.0) and (12; 0), where the speed is set in meters per second, time - in seconds. How many times does the average ground speed for 16 seconds of movement exceed the value of the average speed of movement for the same time?
Solution. The graph of the dependence of the projection of the body's velocity on time is shown in the figure.
To graphically calculate the path traversed by the material point and the modulus of its movement, it is necessary to determine the value of the projection of the velocity at the moment of time, equal to 16 s.
There are two ways to determine the value of v x at a specified moment in time: analytical (through the equation of a straight line) and graphical (through the similarity of triangles). To find v x, we will use the first method and compose the equation of a straight line at two points:
t - t 1 t 2 - t 1 = v x - v x 1 v x 2 - v x 1,
where (t 1; v x 1) - coordinates of the first point; (t 2; v x 2) - coordinates of the second point. By the condition of the problem: t 1 = 0, v x 1 = 8.0, t 2 = 12, v x 2 = 0. Taking into account the specific values of the coordinates, this equation takes the form:
t - 0 12 - 0 = v x - 8.0 0 - 8.0,
v x = 8.0 - 2 3 t.
At t = 16 s, the velocity projection value is
| v x | = 8 3 m / s.
This value can also be obtained from the similarity of triangles.
- Let's calculate the path traversed by the material point as the sum of the values S 1 and S 2:
S = S 1 + S 2,
where S 1 = 1 2 ⋅ 8.0 ⋅ 12 = 48 m - the path traversed by the material point during the time interval from 0 s to 12 s; S 2 = 1 2 ⋅ (16 - 12) ⋅ | v x | = 1 2 ⋅ 4.0 ⋅ 8 3 = = 16 3 m - the path traversed by the material point in the time interval from 12 s to 16 s.
The total distance traveled is
S = S 1 + S 2 = 48 + 16 3 = 160 3 m.
The average ground speed of a material point is
v s = S t 2 - t 1 = 160 3 ⋅ 16 = 10 3 m / s.
- We calculate the value of the displacement of a material point as the modulus of the difference between the values S 1 and S 2:
S = | S 1 - S 2 | = | 48 - 16 3 | = 128 3 m.
The value of the average movement speed is
| v → r | = | Δ r → | t 2 - t 1 = 128 3 ⋅ 16 = 8 3 m / s.
The sought speed ratio is
v s | v → r | = 10 3 ⋅ 3 8 = 10 8 = 1.25.
The average ground speed of a material point is 1.25 times the modulus of the average speed of movement.
If a material point is in motion, then its coordinates are subject to changes. This process can be fast or slow.
Definition 1
The quantity 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 of time, and co-directional with the displacement vector υ = ∆ r ∆ t; υ ∆ r.
Picture 1 . Average speed is in the same direction as movement
The modulus of the average speed along the way is υ = S ∆ t.
Instantaneous speed characterizes movement at a certain point in time. The expression "body speed at a given time" is considered not correct, but applicable in mathematical calculations.
Definition 3
The instantaneous speed is called 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 tangential 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 available 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 = υ = υ x 2 + υ y 2 + υ z 2 = x 2 + y 2 + z 2.
To pass from Cartesian rectangular coordinates to curvilinear ones, the rules for differentiating complex functions are applied. If the radius vector r is a function of curvilinear coordinates r = r q 1, q 2, q 3, then the velocity value will be 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, assume that q 1 = r; q 2 = φ; q 3 = θ, then we get υ presented in the following 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 = r 1 + φ 2 sin 2 θ + θ 2.
Definition 4
Instant speed is the value of the derivative of the time displacement function at a given moment associated with elementary displacement by the relation d r = υ (t) d t
Example 1
A law of rectilinear motion of a point x (t) = 0, 15 t 2 - 2 t + 8 is given. Determine its instantaneous speed 10 seconds after the start of movement.
Solution
It is customary to call the instantaneous velocity the first derivative of the radius vector with respect to time. Then its record will take the form:
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 o s t, when the point stops moving, and its average ground speed υ.
Solution
Let's calculate the equation of the instantaneous velocity, substitute the numerical expressions:
υ (t) = x ˙ (t) = 4 - 0, 1 t.
4 - 0, 1 t = 0; t about with t = 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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