Ballistic Pendulum: Determining Projectile Velocity Through Energy Conservation

[hawk320]

Homework Statement

A ballistic pendulum is a device used to measure the velocity of a projectile. The projectile is shot horizontally into and becomes embedded in the bob of a pendulum as illustrated below. The pendulum swings upward to some height h, which is measured. The mass of the bullet, m, and the mass of the pendulum bob, M, are known. Using the laws of energy and ignoring and rotational considerations, show that the initial velocity of the projectile is given by v_0=(m+M)/m*√2gh

Homework Equations

K=1/2*m*v^2
p=mv
Momentum is conserved
U=mgh

The Attempt at a Solution

I have been trying to manipulate equations into each other, but to no luck. I thought I had something with U+K=U+K, but it didn't simplify to that equation.

 

[Galileo's Ghost]

From Conservation of momentum:
p before collision = p after collision
mv_0 = (m+M)v

since the masses are now combined they will start the pendulum swing upwards with a common velocity v. At the start of this swing motion, all of the energy is in the form of kinetic, but it will be converted to gravitational potential energy as the pendulum climbs higher and eventually will all be gravitational potential energy at height h. So from conservation of energy we can write:

1/2 (m+M)v^2 = (m+M)gh

Use that equation to get an expression for v. Then substitue that expression for v into the momentum equation from earlier and solve for v_0

 

[hawk320]

Thank you for your help.