Average and instantaneous acceleration and speed. Formulas Task example

In physics, kinematics is concerned with the analysis of the motion of macroscopic solids. This section of mechanics deals with concepts such as speed, acceleration, and path. In this article, we will focus on issues such as instant acceleration and speed. We also consider what formulas can be used to determine these quantities.

Finding speed

Every student knows about this concept, starting from the elementary grades. All students are familiar with the following formula:

v = S / t.

Here S is the path that a moving body has covered in time t. This expression allows us to calculate some average velocity v. Indeed, we don’t know how the body moved, on which part of the path it moved faster, and on which slower. Even the situation is not excluded that at some point on the path it was at rest for some time. The only thing that is known is the distance traveled and the corresponding time period.

In high school, speed, as a physical quantity, is seen in a new light. Students are offered the following definition:

v = dS / dt.

To understand this expression, you need to know how the derivative of a function is calculated. In this case, it is S (t). Since the derivative characterizes the behavior of the curve at this particular point, the speed calculated by the formula above is called instantaneous.

Acceleration

If the mechanical motion is variable, then for its accurate description it is necessary to know not only speed, but also a quantity that shows how it changes over time. This is the acceleration, which is the time derivative of the speed. And that, in turn, is a time derivative of the path. The formula for instant acceleration is:

a = dv / dt.

Due to this equality, it is possible to determine the change in v at any point on the trajectory.

By analogy with speed, the average acceleration is calculated by the following formula:

a = Δv / Δt.

Here Δv is the change in the module of the velocity of the body over the period of time Δt. It is obvious that during this period the body is able to both accelerate and slow down. The value of a, determined from the expression above, will show only on average the speed of the change in speed.

Constant Acceleration

A distinctive feature of this type of movement of bodies in space is the constancy of the quantity a, that is, a = const.

This motion is also called uniformly accelerated or equally slow depending on the mutual direction of the velocity and acceleration vectors. Below we consider such a movement using two of the most common trajectories as an example: a straight line and a circle.

When moving in a straight line during uniformly accelerated motion, the instantaneous speed and acceleration, as well as the distance traveled, are related by the following equalities:

v = v ₀ ± a * t;

S = v ₀ * t ± a * t 2/2.

Here v ₀ is the velocity value that the body possessed before the acceleration a appeared. Note one caveat. For this type of movement, it makes no sense to talk about instant acceleration, because at any point on the trajectory it will be the same. In other words, its instantaneous and average values will be equal to each other.

As for speed, the first expression allows you to determine it at any time. That is, it will be an instant indicator. To calculate the average speed, you must use the above expression, that is:

v = S / t = v ₀ ± a * (t ₁ + t ₂ ) / 2.

Here t ₁ and t ₂ are time instants between which the average speed is calculated.

The plus sign in all formulas corresponds to accelerated movement. Accordingly, the minus sign is in slow motion.

In the study of circular motion with constant acceleration in physics, angular characteristics are used that are similar to the corresponding linear ones. These include the angle of rotation θ, angular velocity and acceleration (ω and α). These values ​​are related in equalities, similar to expressions of uniformly accelerated motion in a straight line, which are given below:

ω = ω ₀ ± α * t;

θ = ω ₀ * t ± α * t 2/2.

In this case, the angular characteristics are associated with linear as follows:

S = θ * R;

v = ω * R;

a = α * R.

Here R is the radius of the circle.

The task of determining the average and instantaneous acceleration

It is known that the body moves along a complex path. Its instantaneous speed varies in time as follows:

v = 10 - 3 * t + t ³ .

What is the instantaneous acceleration of the body at time t = 3 (seconds)? Find the average acceleration over a period of time from two to four seconds.

It is not difficult to answer the first question of the problem if we calculate the derivative of the function v (t). We get:

a = | dv / dt | _{t is 2;}

a = | 3 * t ² - 3 | _{t = 2} = 24 m / s ² .

To determine the average acceleration, you should use this expression:

a = (v ₂ - v ₁ ) / (t ₂ - t ₁ );

a = ((10 - 3 * 4 + 4 ³ ) - (10 - 3 * 2 + 2 ³ )) / 2 = 25 m / s ² .

From the calculations it follows that the average acceleration slightly exceeds the instantaneous in the middle of the considered time period.