A rifle has a mass of 4.15 kg and it fires a bullet of mass 13.3 g at a muzzle speed of 823 m/s. What is the recoil speed of the rifle as the bullet leaves the gun barrel?

To solve this problem, we need to use the principle of conservation of momentum. The total momentum before the bullet is fired is zero (since both the rifle and bullet are initially at rest), and the total momentum after the bullet is fired must also be zero (since there are no external forces).

The momentum of an object is given by its mass times its velocity. So, the momentum of the bullet after it is fired is its mass times its velocity, and the momentum of the rifle is its mass times its recoil speed.

Let's denote the recoil speed of the rifle as Vr. We can set up the following equation:

0 = momentum of bullet + momentum of rifle 0 = (mass of bullet * velocity of bullet) + (mass of rifle * Vr)

We can rearrange this equation to solve for Vr:

Vr = - (mass of bullet * velocity of bullet) / mass of rifle

Now we just need to plug in the given values. Note that the mass of the bullet needs to be converted from grams to kilograms:

Vr = - (0.0133 kg * 823 m/s) / 4.15 kg Vr = -10.9 m/s

The negative sign indicates that the rifle moves in the opposite direction of the bullet, which is what we would expect for a recoil. So, the recoil speed of the rifle is 10.9 m/s.