Question

A body of mass $50\,kg$ is projected vertically upwards with velocity of $100\,m{s^{ - 1}}$. $5$ seconds after this body breaks into $20\,kg$ and $30\,kg$. If $20\,kg$ piece travels upwards with $150m{s^{ - 1}}$, then the velocity of other block will be
A. $15\,m{s^{ - 1}}$ downwards
B. $15\,m{s^{ - 1}}$ upwards
C. $51\,m{s^{ - 1}}$downwards
D. $51\,m{s^{ - 1}}$ upwards

Answer 1

Hint:In order to solve this question, we will apply the law of conservation of linear momentum and then using this principle and equations we will determine the velocity of the other block.

Formula used:
The principle of conservation of linear momentum says that Initial momentum of a system is always equal to final momentum of a system.
${P_i} = {P_f}$
where $P = mv$ denotes the momentum of a body defined as the product of mass of the body and the velocity of the body.

Complete step by step solution:
According to the question, we have given that initial velocity of body is $u = 10\,m{s^{ - 1}}$ and after $t = 5s$ final velocity of body will be,
$v = u - gt \\
\Rightarrow v = 100 - 9.8(5) \\
\Rightarrow v = 51\,m{s^{ - 1}} $
Now, just before breaking the body of mass $m = 50\,kg$ and velocity $v = 51\,m{s^{ - 1}}$.
So, initial momentum of the system is,
${P_i} = mv = 2550\,kgm{s^{ - 1}} \to (i)$

Now, after breaking of body in two pieces having mass and velocity as ${m_1} = 20kg,{v_1} = 150\,m{s^{ - 1}}$ and ${m_2} = 30kg,{v_2}$.
So, final momentum of the body will be,
${P_f} = {m_1}{v_1} + {m_2}{v_2} \\
\Rightarrow {P_f} = 3000 + 30{v_2} \to (ii) \\ $
Now, equate equations (i) and (ii) using the law of conservation of linear momentum.
$2550 = 3000 + 30{v_2} \\
\therefore {v_2} = - 15\,m{s^{ - 1}}$
So, a negative sign implies the direction of the piece is downwards.

Hence, the correct answer is option A.

Note: It should be remembered that in the upward motion of the body we have taken a positive direction so, the negative sign implies the opposite direction and here acceleration due to gravity is taken as $g = 9.8\,m{s^{ - 2}}$.