INSTANT ACCELERATION: WHAT IT IS, HOW IT IS CALCULATED AND EXERCISES - PHYSICAL - 2025

The instantaneous acceleration is the change that speed per unit time at each instant of the movement. At the precise moment that the dragster in the image was photographed, it had an acceleration of 29.4 m / s ². This means that at that moment, its speed was being increased by 29.4 m / s in the span of 1 s. This is equivalent to 105 km / h in just 1 second.

A dragster competition is easily modeled by assuming that the racing car is a point object P moving in a straight line. On that line we choose an axis oriented with origin O that we will call the (OX) axis or simply the x axis.

Dragsters are cars capable of enormous accelerations. Source: Pixabay.com

The kinematic variables that define and describe the movement are:

• Position x
• The displacement Δx
• Speed ​​v
• Acceleration to

They are all vector quantities. Therefore they have a magnitude, a direction and a sense.

In the case of rectilinear motion there are only two possible directions: positive (+) in the direction of (OX) or negative (-) in the opposite direction of (OX). Therefore, it is possible to dispense with the formal vector notation and use the signs to indicate the sense of the magnitude.

How is acceleration calculated?

Suppose that at instant t the particle has velocity v (t) and at instant t 'its velocity is v (t').

Then the change that the velocity had in that period of time was Δ v = v (t ') - v (t). Therefore, the acceleration in the time period Δ t = t '- t, would be given by the quotient:

This quotient is the average acceleration a _m in the time Δt between instants t and t '.

If we wanted to calculate the acceleration just at time t, then t 'would have to be a negligibly greater quantity than t. With this Δt, which is the difference between the two, should be almost zero.

Mathematically it is indicated as follows: Δt → 0 and it is obtained:

Solved exercises

Exercise 1

The acceleration of a particle moving along the X axis is a (t) = ¼ t ². Where t is measured in seconds and in m / s. Determine the acceleration and velocity of the particle at 2 s of motion, knowing that at the initial instant t ₀ = 0 it was at rest.

Reply

At 2 s the acceleration is 1 m / s ² and the velocity for time t will be given by:

Exercise 2

An object moves along the X axis with a velocity in m / s, given by:

v (t) = 3 t ² - 2 t, where t is measured in seconds. Determine the acceleration at times: 0s, 1s, 3s.

Answers

Taking the derivative of v (t) with respect to t, the acceleration is obtained at any instant:

a (t) = 6t -2

Then a (0) = -2 m / s ²; a (1) = 4 m / s ²; a (3) = 16 m / s ².

Exercise 3

A metal sphere is released from the top of a building. Falling acceleration is the acceleration of gravity that can be approximated by the value 10 m / s2 and pointing downward. Determine the speed of the sphere 3 s after it has been released.

Reply

This problem involves the acceleration of gravity. Taking the vertical downward direction as positive, we have that the acceleration of the sphere is:

a (t) = 10 m / s ²

And the speed will be given by:

Exercise 4

A metal sphere is shot upward with an initial speed of 30 m / s. The acceleration of motion is the acceleration of gravity that can be approximated by the value 10 m / s ² and pointing downward. Determine the speed of the sphere at 2 s and 4 s after it has been shot.

Reply

The vertical upward direction will be taken as positive. In that case the acceleration of the motion will be given by

a (t) = -10 m / s ²

The speed as a function of time will be given by:

After 4 s of having been fired, the velocity will be 30 - 10 ∙ 4 = -10 m / s. This means that at 4 s the sphere is descending with a speed of 10 m / s.

References

1. Giancoli, D. Physics. Principles with Applications. 6th Edition. Prentice Hall. 25-27.
2. Resnick, R. (1999). Physical. Volume 1. Third edition in Spanish. Mexico. Compañía Editorial Continental SA de CV 22-27.
3. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7th. Edition. Mexico. Cengage Learning Editors. 25-30.