Question

Derive expression for recoil velocity of gun

Answer 1

When a bullet is fired from the gun, due to reaction force , the gun recoils in the direction opposite to the motion of bullet. The  velocity with which it recoils is called as Recoil Velocity.

Derivation For recoil velocity of Gun:

Initial velocity of gun and bullet are at rest.
Hence u1 and u2 =0
let m and M be the mass of bullet and gun respectively.
The sum of momentum before firing bullet is zero.
mu1+Mu2=0If v1 and v2 are the final velcoities of bullet and gun receptively after firing.
Then According to law of conservation of momentum
Total momentum before recoil = Total momentum after recoil.
0=mv1+Mv2
Therefore,
Mv2=-mV1V2
= -mV1/M
This the recoil velocity of Gun.
Here negative sign indicates that the direction of velocity of the gun is opposite to the direction of velocity of bullet.

Answer 2

Law of conservation of momentum definition



According to this law:”The momentum of an isolated system of two or more than two interacting bodies remains constant.”Momentum of a system depends on its mass and velocity.A system is a group of bodies within certain boundaries .An isolated system is a group of interacting bodies on which no external force is acting.It is expressed mathematically as:



Above formula states that momentum of two striking bodies before and after collision remains same.

Conservation of momentum examples in real life

Consider the example of air-filled balloon as described under the third law of motion.In this case, balloon and the air inside it form a system.Before releasing the balloon,the system was at rest and hence the initial momentum of the system was zero.As soon as the balloon is set free,air escapes out of it possesses momentum.To conserve momentum, the balloon moves in a direction opposite to that of air rushing out.

Similarly:

 A system of gun and a bullet.Before firing the gun, both the gun and the bullet are at rest,so the total momentum of the system is zero.As the gun is fired, bullet shoots out of the gun and acquires momentum.To conserve momentum of the system,the gun recoils.According to the law of conservation of momentum,the total momentum of the gun and the bullet will be zero after the gun is fired.Let m be the mass of the bullet and v be the velocity on firing the gun; M be the mass of the gun and V be the velocity with which it recoils.Thus the total momentum of the gun and the bullet after the gun is fired will be:

Total momentum of gun and bullet after the gun is fired = M v + m v ….(1)

According to the  conservation of momentum law:

Total momentum of gun and bullet after the gun is fired =Total momentum of gun and bullet before the gun is fired

Therefore we get:

M V +mv = 0

or M V = -m v

Hence:



Above equation gives the velocity V of the gun, the negative sign indicates that velocity of the gun is opposite to the velocity of the bullet. i .e the gun recoils.Since the mass of the gun is much larger than the bullet, therefore, the recoil is much smaller than the velocity of the bullet.

Example:

A bullet of mass 20 g is fired from a gun with a muzzle velocity 100 ms-1.The mass of the gun is 5 kg.Then the recoil of the gun will be – 0.4 ms-1.The negative sign indicates that the gun recoils i.e moves in the backward direction opposite to the motion of the bullet with a velocity of 4 ms-1.

Rockets and jet engines also work on the same principle.In these machines, hot gases produced by burning of fuel rush out with large momentum.The machines gain an equal and opposite momentum.This enables them to move with very high velocities.