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Question: A particle moves with velocity \(v_1\) for time \(t_1\) and \(v_2\) for time \(t_2\) along a straigh...

A particle moves with velocity v1v_1 for time t1t_1 and v2v_2 for time t2t_2 along a straight line. What is the magnitude of its average acceleration?
A. v2−v1t1−t2\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{1}}-{{t}_{2}}}
B. v2−v1t1+t2\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{1}}+{{t}_{2}}}
C. v2−v1t2−t1\dfrac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}
D. v1+v2t1−t2\dfrac{{{v}_{1}}+{{v}_{2}}}{{{t}_{1}}-{{t}_{2}}}

Explanation

Solution

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=Δvta=\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 v1{{v}_{1}} which changed to velocity v2{{v}_{2}} after a time period t1{{t}_{1}}.
Therefore, change in velocity = final velocity – initial velocity
That is, Δv=v2−v1\Delta v=v_2-v_1
The average acceleration of the particle is written as, a=Δvta=\dfrac{\Delta v}{t} where t is the total time elapsed when acceleration is being calculated. Here, t=t1+t2t={{t}_{1}}+{{t}_{2}}. Substituting the values, we have
a=v2−v1t1+t2a=\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 aa and has SI unit m/s2m/{{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.