Question

A particle moves with velocity $v_1$ for time $t_1$ and $v_2$ for time $t_2$ along a straight line. What is the magnitude of its average acceleration?
A. $\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{1}}-{{t}_{2}}}$
B. $\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{1}}+{{t}_{2}}}$
C. $\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}$
D. $\dfrac{{{v}_{1}}+{{v}_{2}}}{{{t}_{1}}-{{t}_{2}}}$

Answer 1

Hint: Average acceleration of a particle is defined as the ratio of the change in velocity with respect to elapsed time. As particles move along a straight line, the change in velocity can be calculated directly by taking differences in the velocities. The time elapsed is the sum of the time the particle moves with constant velocity.
Formula used:
Average acceleration, $a=\dfrac{\Delta v}{t}$

Complete step by step answer:
Average acceleration of a particle is defined as the ratio of the change in velocity with respect to elapsed time. As particles move along a straight line, the change in velocity can be calculated directly by taking differences in the velocities. Initially, the particle was moving with velocity ${{v}_{1}}$ which changed to velocity ${{v}_{2}}$ after a time period ${{t}_{1}}$.
Therefore, change in velocity = final velocity – initial velocity
That is, $\Delta v=v_2-v_1$
The average acceleration of the particle is written as, $a=\dfrac{\Delta v}{t}$ where t is the total time elapsed when acceleration is being calculated. Here, $t={{t}_{1}}+{{t}_{2}}$. Substituting the values, we have
$a=\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{1}}+{{t}_{2}}}$
The obtained value of acceleration matches with option.

So, the correct answer is “Option B”.

Additional Information:
Acceleration of a particle is defined as the rate of change of velocity. It is denoted by $a$ and has SI unit $m/{{s}^{2}}$. Unlike acceleration, the average acceleration is calculated for a given interval.

Note:
Average acceleration of a particle is the ratio of the change in velocity with respect to elapsed time. When a particle moves along a straight line or say in one direction only, the vector difference is the same as the difference between the magnitudes of the quantities.