Question

A gun of mass $M$ fires a bullet of mass $m$ with maximum speed $v$. Given that $m<M$ what is the kinetic energy of the gun?

Answer 1

Step 1. Given data

Mass of the gun $=$$M$

Mass of the bullet $=$ $m$

Maximum speed $=$$v$

Step 2. Formula used

Kinetic energy $\mathrm{K}.\mathrm{E}=\frac{1}{2}m{v}^2$

Momentum of a body $p=mv$

where, $m$ is the mass and $v$ is the velocity

Step 3. Calculate the velocity of the gun

A gun recoils when a bullet is discharged from it. Both the pistol and the bullet travel in the polar opposite directions. In this instance, the gun's linear momentum and the bullet's linear momentum are equal. Since mass and velocity have an inversely proportional relationship, the mass of the gun is very big compared to the mass of the bullet, and the rifle's recoil velocity is very slow compared to the forward velocity of the bullet.

$m_1=$Mass of bullet

$m_2=$Mass of gun

$\overrightarrow{v_1}=$velocity of bullet

$\overrightarrow{v_2}=$Velocity of the gun

Since the gun and bullet are both at rest before to firing, their combined linear momentum is zero. Following firing, the total linear momentum is given by

$$\begin{gathered} \therefore m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}=0 \\ \Rightarrow \overrightarrow{v_2}=\left(\frac{m_1}{m_2}\right)\left(-\overrightarrow{v_1}\right) \end{gathered}$$

The negative sign in the above equation shows that the direction of $\overrightarrow{v_2}$ is exactly opposite of $\overrightarrow{v_1}$.

Mass of the gun $=$$M$

Mass of the bullet $=$ $m$

Velocity of bullet$=$$v$

Velocity of Gun$=V$

Now applying the law of conservation we have

$\Rightarrow MV=mv$

$\Rightarrow V=\frac{mv}{M}$

Step 4. Calculate the kinetic energy of the gun

The kinetic energy of the gun is

$$\begin{gathered} \mathrm{K}.\mathrm{E}=\frac{1}{2}M{V}^2 \\ =\frac{1}{2}\times \mathrm{M}\times {\left(\frac{\mathrm{mv}}{\mathrm{M}}\right)}^2 \\ =\frac{1}{2M}{\left(mv\right)}^2 \end{gathered}$$

Hence, the kinetic energy of the gun is $\frac{1}{2M}{\left(mv\right)}^2$.