Question

A man weighing $80kg$ is standing on a trolley weighing $320kg$. The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley along the rails at the speed $1m{{s}^{-1}}$. Then after $4s$, his displacement relative to the ground will be:
a) $4.5m$
b) $5m$
c) $8m$
d) $3.2m$

Answer 1

Hint : When the man moves forward he applies a normal force on the trolley which makes it move backward. Some of the momentum from his foot gets transferred to the trolley. Hence we can use the conservation of momentum to calculate their final velocity.

Complete solution:
In the above case the man moves with some velocity and hence has some momentum. This momentum gets transferred to the trolley due the normal produced by the trolley due to a backward force by the foot.
Using law of conservation of momentum,
The forward momentum of the man = backward motion of the man and the trolley moving together.
Since momentum of an object is given by the product of mass times the velocity, i.e. mathematically given by p= mv $kgm{{s}^{-1}}$.
Mathematically we can represent the above condition as,
${{P}_{FORWARD}}={{P}_{BACKWARD}}$
Mass of the man is $80kg$ and moves with velocity of $1m{{s}^{-1}}$
Mass of man + Mass of trolley=$320+80=400kg$
Substituting in the above equation conservation of momentum,
${{P}_{FORWARD}}={{P}_{BACKWARD}}$
$80\times 1=400\times v$ where v is the velocity of trolley moving together backwards
$v=0.2m{{s}^{-1}}$
For every 1 sec when the man moves ahead on the trolley, the trolley moves back by $0.2m$.
Hence in four seconds the man will move $4$ m but since the trolley moves back by $4\times 0.2=0.8m$
The displacement of man with respect to ground will be $4-0.8=3.2m$

Hence, the correct answer to the above question is option c.

Note:
The relative displacement with respect to ground is considered by considering the velocity of the other objects moving relative to you as zero. This condition is only possible in an inertial frame. Inertial frame is a non accelerating frame where no object is under acceleration.