A bullet of mass 44 grams and moving at unknown velocity is quickly absorbed into the 3.8 kg mass of a ballistic pendulum, initially at rest, which quickly and completely absorbs the bullet. The mass of the pendulum is observed to move to a maximum displacement of .085 meters from the equilibrium position. From the length and mass of the pendulum we determine that the restoring force constant is k = m g / L.
At the extreme position, 44 meters from equilibrium, the potential energy of the mass is .5 kx ^ 2 = 8.139 Joules. Since only the restoring forces were acting on the mass between that time and the time the extreme position was attained, no work was done on the system between the instant after the collision and the time at which the maximum displacement was attained. Thus no energy was added to the system during this time.
Therefore the 8.139 Joules of energy must have been present in the form of kinetic energy just after collision.
It follows that, just after collision, the total mass 3.844 of the bullet and the larger mass had kinetic energy
initial total KE = 8.139 Joules.
Solving for the velocity we obtain
v = 4.235 meters/second.
The mass therefore has momentum
momentum = ( 3.844 kilograms) ( 4.235 meters/second) = 16.27 kilogram meters/second.
Since momentum is conserved in collisions, the total momentum before collision was also 16.27 kilogram meters/second. Before the collision, only the bullet had momentum. If we let v stand for its velocity for collision, we therefore have
( 44 grams)* vBullet = ( .044 kilograms) * vBullet = 16.27 kilogram meters/second.
Solving for vBullet, we obtain the bullet's velocity
"vBullet = 370 meters/second.