Question

A machine gun fires n bullets per second and the mass of each bullet is m. If v is the speed of each bullet, then the average force exerted on the machine gun is​

Answer 1

⇝ Given :-

• A machine gun fires n bullets per second

• The mass of each bullet is m

• The Velocity of each bullet is v

⇝ To Find :-

• The average force exerted on the machine gun.

⇝ Solution :-

Here,

• Initial Velocity if each bullet = 0

• Final Velocity of each bullet = v

• Mass of each bullet = m

❒ We Know that Change in Momentum of any Body is :

$\large \bf \red\bigstar \: \: \orange{ \underbrace{ \underline{ \Delta P = m(v_f-v_i)}}}$

Where,

• $\text{m = Mass of Body}$

• $\text v_ \text i = \text{Initial Velocity of Body}$

• $\text v_ \text f = \text{Final Velocity of Body}$

Therefore,

✧ Change in Momentum for One bullet is :

$\rm\Delta P_1 = m(v - 0)$

$\large:\longmapsto \pmb{\boxed{ \rm \Delta P_1 = mv}}$

✧ According To Question Number of bullets fired in one second is n

Hence,

Change in Momentum for bullets fired in one second is :

$\rm \Delta P = n(Change \: in \: Momentum \: for \: One \: bullet)$

$:\longmapsto \rm \Delta P = n(\Delta P_1)$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf \Delta P = mnv} }}}$

❒ We Know that relationship between Force and change in momentum for any body is :

$\large \bf \red\bigstar \: \: \orange{ \underbrace{ \underline{F= \frac{\Delta P}{\Delta t} }}}$

Therefore,

Force Exerted on Machine Gun is :

$\rm F = \frac{\Delta P}{1}$

$:\longmapsto \rm F = \Delta P$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf F = mnv} }}}$

Hence,

$\large\underline{\pink{\underline{\frak{\pmb{ Force \: Exerted = mn \text v }}}}}$

Answer 2

Assuming,

Initial velocity of the bullet, u=0

Final velocity of the bullet is v

Therefore,

Change in momentum of one bullet is: ΔP1

So, change in momentum is equal to: ΔP1=mv−0=mv

In one second, the number of bullets fired is n.

So change in momentum of the bullets in one second is equal to: ΔP=nΔP1=mnv

Therefore,

According to the definition of speed.

Force exerted on the machine gun is equal to; F=ΔPΔt

On putting in the values we get.

F=mnv1=mnv