Why does a gun recoil? Derive the recoil velocity of the gun.

A gun recoils due to the principle of conservation of momentum. When a bullet is fired, the gun experiences an equal and opposite reaction to the bullet's forward motion, resulting in the gun moving backward. This phenomenon is described by Newton's third law of motion, which states that for every action, there is an equal and opposite reaction.

Understanding Recoil

When the bullet is discharged, it gains forward momentum. To maintain the overall momentum of the system (gun plus bullet), the gun must move backward. The amount of recoil depends on the mass of the bullet and its velocity at the moment of firing.

Calculating Recoil Velocity

The recoil velocity of the gun can be derived using the following formula:

• m_b: mass of the bullet
• v_b: velocity of the bullet
• m_g: mass of the gun
• v_g: recoil velocity of the gun

The conservation of momentum states that the total momentum before firing equals the total momentum after firing. Initially, both the gun and bullet are at rest, so:

0 = m_b * v_b + m_g * (-v_g)

Rearranging this gives:

m_b * v_b = m_g * v_g

From this, we can solve for the recoil velocity of the gun:

v_g = (m_b * v_b) / m_g

Example Calculation

For instance, if a bullet has a mass of 0.01 kg (10 grams) and a velocity of 400 m/s, and the gun has a mass of 2 kg, the recoil velocity can be calculated as follows:

v_g = (0.01 kg * 400 m/s) / 2 kg = 2 m/s

This means the gun will recoil backward at a velocity of 2 m/s when the bullet is fired.