Question

A trolley is moving horizontally with a constant velocity of v with respect to earth.
A man starts running from one end of the trolley with a velocity 1.5v with respect to the trolley. After reaching the opposite end, the man returns back and continues running with velocity 1.5v w.r.t the trolley in the backward direction. If the length of the trolley is L, then the displacement of the man with respect to earth during the process will be:-

A. 2.5 L
B. 1.5 L
C. \[\dfrac{{5L}}{3}\]
D. \[\dfrac{{4L}}{3}\]

Answer 1

Hint:Determine the time taken by the man to cover the distance 2L, where, L is the length of the trolley. Since the trolley moves with velocity v with respect to the earth, the displacement of the man will be the product of velocity of the trolley w.r.t the earth and the time taken by the man to cover 2L distance.
Formula used:
Velocity,\[v = \dfrac{d}{t}\]
Here, d is the displacement and t is the time.

Complete step by step answer:
We have given that the velocity of the trolley is v with respect to earth and velocity of the
man is 1.5v with respect to earth. The length of the trolley is L and therefore, the total distance covered by the man is 2L.
Let’s determine the time taken by the man to cover a total distance of 2L as follows,
\[t = \dfrac{{2L}}{{1.5v}}\]
We have given that the trolley is also moving with a velocity v with respect to earth.
Therefore, the displacement of the man with respect to the earth will be the product of the
velocity of the trolley with respect to the earth and the time taken by the man to cover 2L
distance. Therefore, we can write,
\[d = v \times \dfrac{{2L}}{{1.5v}}\]
\[ \Rightarrow d = \dfrac{2}{{\left( {3/2} \right)}}L\]
\[ \Rightarrow d = \dfrac{4}{3}L\]
So, the correct answer is option (D).

Note: Students must note that the displacement of the man with respect to the trolley is zero. The man covers a maximum displacement in the forward direction because the man and the trolley move in the same direction and therefore the velocity of the man will be the sum of velocity of the man with respect to the trolley and velocity of the trolley with respect to the earth.