Question

A person of mass $m$ is standing on one end of a plank of mass $M$ and length $L$ and floating in the water. The person moves from one end to another and stops. The displacement of the plank is

• A. $\frac{Lm}{(m+M)}$
• B. $\frac{L(M+m)}{m}$
• C. $\frac{(M+m)}{Lm}$
• D. $\frac{LM}{(m+M)}$

Answer 1

The correct option is A

$\frac{Lm}{(m+M)}$

Step 1: Given data

Mass of the person$=m$

Mass of the plank $=M$

Length of the plank $=L$

The diagrammatic representation of the given data is:

Step 2: Formula used

Conservation of momentum

Initial momentum$=$Final momentum

Step 3: Solution

Let the displacement of the plank is $x$, the velocity of the person with respect to the plank be $v_1$and the velocity of plank is opposite direction is $v_2$

According to the law of conservation of momentum

$m{v}_1=(m+M)v_2$

$$\begin{gathered} \Rightarrow m\frac{L}{t}=(m+M)\left(\frac{x}{t}\right) \\ \Rightarrow mL=(m+M)\left(x\right) \\ \Rightarrow x=\frac{mL}{m+M} \end{gathered}$$

Hence, the correct option is $A$.