Question

You work for the National Park Service testing a small cannon used to prevent avalanches by shooting down snow overhanging the sides of mountains. In order to determine the range of the cannon, it is necessary to know the speed with which the projectile leaves the cannon (muzzle speed), relative to the ground. The cannon you are testing has a weight of 500 lbs. and shoots a 20-lb. projectile. During lab tests where the cannon is held and cannot move, the muzzle speed is 400 m/s. You want to calculate the projectile's muzzle speed with respect to the ground under field conditions when the cannon is mounted so that it is free to move (recoil) when fired. You take the case where the cannon is fired horizontally using the same shells as in the laboratory.

Answer 1

When the cannon is fixed the whole energy is given to the bullet, but when the cannon is not fixed, a part of the energy is gone to cannon too, since it also moves due to recoil. The energy produced in both cases are the same, from that we can write,

(in the initial case)

Substituting values we get,

(in the case when recoil is there)

Substituting values we get,

By conservation of energy, we will get the equation,

Also from conservation of momentum, we can write,

Substituting values we get,

Solving the above two equations, we get, the bullet and cannon velocities as,

(we know that the bullet velocity will be positive and the cannon velocity will be negative, hence only considering such a solution).

Hence the muzzle speed at the field is 392.23 m/s.