# Conservation of Momentum

If a metal ball strikes a pendulum and sticks to it. It will have a relationship for conservation of momentum of m*v = (m + M) *V (m and v are the mass and initial velocity of the projectile, M is the mass of the pendulum with no initial velocity and big V is the combined velocity of the pendulum and projectile).

The conservation of energy after the collision is :

1/2 (m + M) * V^2 = (m + M)gh

The vertical rise of the pendulum is represented by h and the horizontal distance swung through is d. The quantities are related by:

R^2 = (R-h)^2 + d^2

Based on these equations, I need to derive a relationship for little v (on quantities m,M, g, R, d) and for d (in terms of m, M, g, R, and v)but am having some difficulty getting this to work out nicely. Any guidance?

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#### Solution Preview

m*v = (m + M) *V

or v= (m + M) *V/ m --------------------(1)

Â½ (m + M) * V^2 = (m + M) g h

Canceling (m + M) from both sides and rearranging

V^2 = 2 g h or V= (2 g h) ^ Â½ --------------------(2)

R^2 = (R-h)^2 + d^2

Expanding

R^2 = ...

#### Solution Summary

The solution derives an equation for a metal ball striking a pendulum and sticking to it.