Question

A strip of wood of mass $M$ and length $l$ is placed on a smooth horizontal surface. An insect of mass $m$ starts at one end of the strip and walks to the other end in time $t$, moving with a constant speed. Then which of the following statement(s) is/are correct :

• A. The speed of the insect as seen from the ground is < $\frac{l}{t}$.
• B. The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{M}{M+m}\right)$.
• C. The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{m}{M+m}\right)$.
• D. The total kinetic energy of the system is $\frac{1}{2}(m+M){\left(\frac{l}{t}\right)}^2$.

Answer 1

Answer: (A), (C)

The speed of the insect as seen from the ground is < $\frac{l}{t}$.
The speed of the strip as seen from the ground is $\frac{l}{t}\left(\frac{m}{M+m}\right)$.

Assume speed of strip is $v_2$ and of insect is $v_1$ respectively w.r.t. ground.

Using momentum conservation:

$m{v}_1=M{v}_2$...............(i)

The relative speed of insect w.r.t. plank is :$v_{12}=v_1-(-v_2)=v_1+v_2$

time taken by insect in walking the length of wooden strip:

$t=\frac{l}{v_1+v_2}$...............(ii)

$v_1+v_2=\frac{l}{t}$..............(iii)

From equation (i) and (iii)

$v_1+\frac{m}{M}{v}_1=\frac{l}{t}$.

$v_1=\frac{M}{m+M}\frac{l}{t}$.

$v_2=\frac{m}{m+M}\frac{l}{t}$.