Question

A field gun of $150\,kg$ fires a shell of mass $1\,kg$ with velocity of $150\,m{s^{ - 1}}$. Calculate the velocity of the recoil of the gun.
(A) $1\,m{s^{ - 1}}$
(B) $3\,m{s^{ - 1}}$
(C) $1.5\,m{s^{ - 1}}$
(D) $5\,m{s^{ - 1}}$

Answer 1

In this question, the mass and velocity of the shell is given and the mass of the gun is given. By using the momentum equation, then the velocity of the gun is determined by equating the momentum equation of the gun with the momentum equation of the shell.

Formulae Used:
The momentum equation is given by,
$p = m \times v$
Where, $p$ is the momentum, $m$ is the mass of the object and $v$ is the velocity of the object.

Complete step-by-step solution:
Given that,
The mass of the gun, ${m_g} = 150\,kg$
The mass of the shell, ${m_s} = 1\,kg$
The velocity of the shell, ${v_s} = 150\,m{s^{ - 1}}$
Now, the momentum of the gun is given by,
${p_g} = {m_g} \times {v_g}\,...............\left( 1 \right)$
Where, ${p_g}$ is the momentum of the gun, ${m_g}$ is the mass of the gun and ${v_g}$ is the velocity of the gun.
Now, the momentum of the shell is given by,
${p_s} = {m_s} \times {v_s}\,...............\left( 2 \right)$
Where, ${p_s}$ is the momentum of the gun, ${m_s}$ is the mass of the gun and ${v_s}$ is the velocity of the gun.
By equating the equation (1) and equation (2), then
${m_g} \times {v_g} = {m_s} \times {v_s}$
By substituting the mass of the gun and mass of the shell and velocity of the shell in the above equation, then the above equation is written as,
$150 \times {v_g} = 1 \times 150$
By keeping the velocity of the gun in one side and the other terms in other side, then the above equation is written as,
${v_g} = \dfrac{{1 \times 150}}{{150}}$
By dividing the above equation, then we get,
${v_g} = 1\,m{s^{ - 1}}$
Thus, the above equation shows the velocity of the recoil of the gun.
Hence, the option (A) is the correct answer.

Note:- The momentum equation of the gun and the momentum equation of the shell is equated by the law of conservation of momentum. The alternate method of this problem is the total momentum of the two objects before collision is equal to the total momentum of the two objects after collision, by this statement also, the velocity of the bullet can be determined.