Question

A boy and a block, both of same mass $m\text{kg}$, are initially at the same horizontal level and attached to a frictionless pulley through a light inextensible string that moves over the frictionless pulley as shown in figure. The boy starts moving upward with an acceleration of $5{\text{m/s}}^2$ relative to the rope. If the block has to travel a total distance of $10\text{m}$ before reaching at the level of pulley, the time taken by the block in doing so is equal to

• A. $\sqrt{8}\text{s}$
• B. $2\sqrt{2}\text{s}$
• C. $\frac{10}{\sqrt{2}}\text{s}$
• D. $8\text{s}$

Answer 1

The correct option is B $2\sqrt{2}\text{s}$

FBD of the boy and the block is given below. Assume upward direction to be positive.


Acceleration of block w.r.t ground $=a_m$

Now, acceleration of boy w.r.t ground will be:

$a_{bg}=a_{br}+a_{rg}=a_{br}-a_m=5-a_m$

For the boy,
$T-mg=m{a}_{bg}=m\times (5-a_m)$ ..(1)

For mass $m$,
$T-mg=m{a}_m$ ... (2)

From equations (1) and (2), we get $a_m=\frac{5}{2}=2.5{\text{m/s}}^2$

From the eqn $S_y=u_{y}t+\frac{1}{2}{a}_m{t}^2$,
Putting $S_y=10m$ and $u_y=0$

$\Rightarrow t=\sqrt{\frac{2\times 10\times 2}{5}}=2\sqrt{2}\text{s}$
Hence option (B) is correct.