Question

A particle moves with velocity \[{v_1}\] for time \[{t_1}\] and \[{v_2}\] for time \[{t_2}\] along a straight line. The magnitude of its average acceleration is
A. \[\dfrac{{{v_2} - {v_1}}}{{{t_1} - {t_2}}}\]
B. \[\dfrac{{{v_2} - {v_1}}}{{{t_1} + {t_2}}}\]
C. \[\dfrac{{{v_2} - {v_1}}}{{{t_1} - {t_2}}}\]
D. \[\dfrac{{{v_2} + {v_1}}}{{{t_1} - {t_2}}}\]

Answer 1

Hint:The terms such as speed and velocity determine how fast or slow a body or an object is moving. Speed can be determined for any direction, but velocity is only for one direction.

Formula Used:
\[a = \dfrac{{\Delta v}}{t}\],
where $a$ is the acceleration, $t$ is the total time and \[\Delta v\] is the relative velocity.

Complete step by step solution:
Acceleration is the relative velocity or the change in velocity with respect to time along a straight line. For a particle moving with velocity \[{v_1}\] for time \[{t_1}\] and \[{v_2}\] for time \[{t_2}\], total time it is moving is \[{t_1} + {t_2}\]. The relative velocity will be given by \[{v_2} - {v_1}\].

Let \[\Delta v = {v_1} - {v_2}\] and \[t = {t_1} + {t_2}\].
The magnitude of its average acceleration is given as,
\[a = \dfrac{{\Delta v}}{t} \\
\Rightarrow a = \dfrac{{{v_1} - {v_2}}}{{{t_1} + {t_2}}} \\ \]
So, for a particle move with velocity \[{v_1}\] for time \[{t_1}\] and \[{v_2}\] for time \[{t_2}\] along a straight line, the magnitude of its average acceleration is \[\dfrac{{{v_2} - {v_1}}}{{{t_1} + {t_2}}}\].

Therefore, option B is our required solution.

Additional Information: The term “relative” in relative velocities refers to the difference in the velocity of both the moving objects and “average” in the average acceleration means finding the relative velocity with respect to time.

Note:For the total time, add the given time and don’t think of it as taking the relative time and subtract them. Only the velocity is relative so subtract the velocities. If the answer comes in negative then it means only the direction is opposite.