A ballistic pendulum is a device used to measure the speed of a projectile, such as a bullet. The projectile, of mass m, is fired into a large block (of wood or other material) of mass M, which is suspended like a pendulum. (Usually M is somewhat greater than m). As a result of the collision, the pendulum and projectile together swing up to a maximum height h. Determine the relationship between the initial horizontal speed of the projectile, v and the maximum height h.

v = ((m + M) / m) * √ (2*g*h)

Explanation:

Given

m = mass of the projectile

M = mass of the ballistic pendulum

v = initial speed of the projectile

v' = speedof the system (pendulum + projectile) after the inelastic collision

h = maximum height reached for the system

Knowing that is an inelastic collision we have

m*v + M * (0) = (m+M) * v'

⇒ v' = m*v / (m+M)

After the collision, we apply the Principle of the Conservation of Energy

Ki + Ui = Kf + Uf

where

Ui = Kf = 0 J

then

Ki = Uf

0.5 * (m+M) * v'² = (m+M) * g*h

⇒ 0.5*v'² = g*h

⇒ v'² = 2*g*h

⇒ (m*v / (m+M)) ² = 2*g*h

⇒ v = ((m+M) / m) * √ (2*g*h)