When a material point moves, its coordinates change. Coordinates can change quickly or slowly.

The physical quantity that characterizes the speed of change of coordinate is speed ().

Average speed

The average speed of movement is a physical quantity equal to the ratio of the point's displacement vector to the time interval during which this displacement has occurred.

The average speed is a quantity numerically equal to the displacement per unit time.

Speed is a vector quantity.

The direction of the mean velocity vector always coincides with the direction of the displacement vector:

If a point moves rectilinearly in one direction, then

Therefore, the average speed modulus along the path is:

In the international system of units (SI), speed is measured in meters per second:

In the GHS unit system (the name of the first letters of the three main units: centimeter, gram, second), the speed is measured in centimeters per second:

The instantaneous speed instant is the speed at a given time.

Instantaneous speed is defined as the limit of the ratio of the displacement vector to the time interval during which this movement occurs, when the time interval tends to zero:

From the point of view of mathematics, formula (3) is the definition of the first time derivative of the radius vector:

The velocity vector, like any vector, can be defined by three components along the coordinate axes:

those. components of the velocity vector are expressed by time derivatives of the corresponding coordinates of the point.

Note. If the form of the functions expressing the dependence of the coordinates on time is known, then we obtain the velocity components by differentiating these functions with respect to time. On the contrary, if it is known how the components of the velocity of a point depend on time, then using the inverse operation — integration — we will find the form of functions expressing the dependence of the coordinates on time (see the note in § 7).

The instantaneous velocity vector is tangent to the path. Based on this, we can give the following definition of the trajectory:

A trajectory is a line tangent to each point of which coincides with the direction of the velocity vector at these points.

By the nature of the change in speed, mechanical movements are classified as uniform and uneven.

With uniform motion, the speed module at any time is a constant value:

| cp | = | mgn | = const | | = const

In case of non-uniform (variable) movement, the speed module changes:

and the formula for the average velocity module (formula (2)):

- A variable motion, in which the speed modulus increases, (v> v 0) is the accelerated motion.

- A variable motion in which the speed modulus decreases (v 0) is slow motion.

1. What speed is called average speed?

2. What speed units can you name?

3. What speed is called instantaneous speed?

4. Does the speed modulus change with uniform movement?

5. Does the speed modulus change with uneven movement?

# How to calculate instantaneous speed

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Speed is the speed of moving an object in a given direction. For general purposes, finding the speed of an object (v) is a simple task: you need to divide the displacement (s) for a certain time (s) by this time (t), that is, use the formula v = s / t. However, in this way an average body speed is obtained. Using some calculations, you can find the speed of the body at any point in the path. This speed is called **instantaneous speed** and is calculated by the formula **v = (ds) / (dt)**, that is, it is a derivative of the formula for calculating the average speed of the body.

### Instant Speed: Calculation Formula

This parameter is equal to the limit (denoted by limit, abbreviated lim) of the ratio of displacement (coordinate difference) to the period of time during which this change occurred, provided that this period of time tends to reach zero. This definition can be written as the following formula:

v = Δs / Δt as Δt → 0 or so v = lim _{Δt → 0} (Δs / Δt)

Note that the instantaneous velocity is a vector quantity. If the movement occurs in a straight line, then it changes only in magnitude, and the direction remains constant. Otherwise, the instantaneous velocity vector is directed tangentially with respect to the trajectory of movement at each point under consideration. What is the meaning of this indicator? Instantaneous speed allows you to find out what kind of movement the object will carry out per unit of time, if from the moment in question it moves uniformly and rectilinearly.

In the case of uniform movement, there are no difficulties: you just need to find the ratio of distance to time over which it was overcome by the object. In this case, the average and instantaneous velocity of the body are equal. If the movement is inconsistent, then in this case it is necessary to find out the magnitude of the acceleration and determine the instantaneous velocity at each particular moment in time. When moving vertically, the effect of gravity acceleration should be taken into account. The vehicle’s instantaneous speed can be detected using a radar or speedometer. It should be borne in mind that the movement in some sections of the path can take a negative value.

## Instantaneous speed with the rectilinear movement of a material point

When considering uneven motion, it is often not the average speed of the body that is interested, but the speed at a certain point in time, or instantaneous speed. So, if the body hit an obstacle, then the force of the body's influence on the obstacle at the moment of impact is determined by the speed at the moment of impact, and not by the average speed of the body. The shape of the trajectory of the projectile and its range depends on the speed at the time of launch, and not on average speed.

The average speed ($ left langle v right rangle $) of a material point along the X axis is:

[ left langle v right rangle = frac < Delta x> < Delta t> left (1 right), ]

$ Delta t $ - time interval of the body movement.

**Instant speed** we define as the limit to which the average speed tends over an infinitely small period of time:

Such a limit is called a derivative in mathematics:

Expression (3) means that the instantaneous speed (speed at a certain point in time) is a derivative of the coordinate. With the rectilinear movement of a material point, the Instantaneous Speed can be defined as the derivative of the path ($ s $) in time:

## Instantaneous speed of uniform movement of a material point

The average speed of a uniformly moving point is a constant value, which means that the instantaneous speed of a uniformly moving point is a constant value.

The speed of uniform motion is numerically equal to the tangent of the angle of inclination of the line to the time axis (Fig. 1):

[v = k tg alpha left (4 right), ]

where $ k $ is a dimensionless coefficient that determines the ratio of the scale of units of displacement (ordinate axis) and units of time (abscissa axis).

In a graphical representation of the variable motion of a material point, the instantaneous speed is numerically equal to the tangent of the angle of inclination of the tangent to the graph and the abscissa.

## Instantaneous curvilinear speed

We set the position of the material point on the trajectory with the radius vector $ overline

speed is a vector directed tangentially to the trajectory of a material point at the location of the particle.

## Examples of tasks with a solution

**The task.** Two material points move according to the equations:

at what point in time will the speeds of these points be equal?

**Decision.** The problem is about finding the time when the instantaneous speeds of material points will be equal. The instantaneous velocity value will be found as:

Then substituting in turn the equations from system (1.1) we obtain:

We equate the right-hand sides of the equations in system (1.3), find the moment in time at which the velocities are equal ($ v_1 = v_2 $):

[- 3 + 8t-3t ^ 2 = 1-4t-3t ^ 2 to 8t + 4t = 1 + 3 to 12t = 4 to t = frac <1> <3> left (c right ). ]

**Answer.** $ t = frac <1> <3> $ s

**The task.** The material point moves on the XOY plane. The law of change of the coordinate $ x $ is given by the graph in Fig. 2. The coordinate $ y $ is given by the analytical expression: $ y = At (1 + Bt) $, where $ A $ and $ B $ are constant values. Write down an expression relating instantaneous speed and time ($ v (t) $).

**Decision.** From fig. 2, we can write an equation that determines the change in the coordinate $ x $ from time to time:

It turned out that the motion of a material point in the XOY plane is described using a system of equations:

Find the components of the velocity of the material point: