Question

A man of mass 60 kg is standing on a boat of mass 140 kg, which is at rest in still water. The man is initially at 20 m from the shore. He starts walking on the boat for 4s with constant speed $1.5m/s$ towards the shore. The final distance of the man from the shore is:
A.) 15.8 m
B.) 4.2 m
C.) 12.6 m
D.) 14.1 m

Answer 1

Hint: In solving this problem try to make use of the concepts of centre of mass of system and the conservation of momentum of a system. First try to find the distance covered by the man towards the shore. Then using the concepts of conservation of momentum find the distance moved by the boat backwards. Then we will find our answer.

Complete step by step answer:
Speed of the man walking on the boat is, $v=1.5m{{s}^{-1}}$

Total time he was walking on the boat. $t=4\operatorname{s}$

Now, the distance covered by the man in the boat is,

$\begin{aligned} & s=vt \\\ & s=1.5m{{s}^{-1}}\times 4s \\\ & s=6m \\\ \end{aligned}$

The man will walk forward on the boat by 6 m. Now, the boat will also displace in the opposite direction to make the position of the centre of mass of the whole system undisturbed.

Suppose, the velocity of the boat in the opposite direction is V and the distance it goes backwards is x.

Now, according to conservation of momentum law, the initial momentum of the system is equal to the final momentum of the system.

$\begin{aligned} & \left( 140+60 \right)\times V=60\times 1.5 \\\ & V=\dfrac{60\times 1.5}{200} \\\ & V=0.45m{{s}^{-1}} \\\ \end{aligned}$

So, velocity of the boat will be $V=0.45m{{s}^{-1}}$
So, the displacement of the boat in the backward direction is given by,

$\begin{aligned} & S=Vt \\\ & S=0.45m{{s}^{-1}}\times 4s \\\ & S=1.8m \\\ \end{aligned}$

The man was initially at a distance of 20 m from the shore. After the man walks 6 m towards the shore the boat moves 1.8 m backwards away from the shore.

So, the final distance of the man towards the shore is,

Distance = $20-s+S=20-6+1.8=15.8m$

The man is 15.8 m away from the shore.

The correct option is (A)

Note: Centre of mass of the system can be defined as the average position weighted according to their masses.
Conservation momentum gives us that the total momentum of a system remains constant. i.e. the initial momentum of a system is equal to the final momentum of the system.