Question

Show a complete derivation of the “muzzle speed” of the bullet from the ballistic pendulum. You will use conservation of energy and conservation of momentum to determine vmuzzle in terms of mB the mass of the bullet, MP the mass of the pendulum, and h the change in height of the combined pendulum/bullet mass after they stick together. This will be called the “ballistic pendulum equation”

Answer 1

Asuume the bullet initially has a muzzle speed u. The bob(mass M) of the pendulum is initially at rest, when the bullet(mass m) strikes the bob, the bob and the pendulum will be moving with a velocity lesser than the initial velocity of the bullet, let us call this vellocity as v.

The total momentum of the bullet-pendulum before collision is,

The total momentum of the bullet-pendulum after collision is,

According to the law of conservation of momentum(as no extrnal force is acting on the system):

ie,

Which implies that the muzzle velocity is given by,

.................(1)

Now let us find an expression for v in terms of the height reached by the pendulum using the principle of conservation of energy.

Let us take the potential energy at the initial height of the bob to be zero, then the total energy of the system when the bulet strikes the bob is given by just its kinetic energy,

When the bob reaches its maximum height the kinetic energy will be zero and it will only have a potential energy given by,

Which implies,

ie,

Substituting this result in (1), we will achieve the equation for muzzle velocity in terms of height achieved by the pendulum.