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**1. Homework Statement**

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A spring whose spring constant is k is suspended from the ceiling. A block of mass M hangs from the spring. A bullet of mass m is fired vertically upward into the bottom of the block. The spring's maximum compression d is measured.

Find an expression for the bullet's speed.

Express your answer in terms of the variables m, M, k, d, and constant g.

**2. Homework Equations**

Ki + Ug + Usp = Kf + Ug + Usp

**3. The Attempt at a Solution**

I used conservation of momentum to find the final velocity of the bullet+block

(m+M)vf=mvi + Mvi

so...

vf=(m/m+M)*vB

where vB is the initial speed of the bullet.

Next, I used:

Ki + Ug + Usp = Kf + Ug + Usp

(1/2)mvi^2 + (1/2)kd^2 + Mgd = (1/2)mvf^2 + (1/2)kd^2 + Mgd

0 + (1/2)kd^2 + 0 = (1/2)(m+M)[(m+M)^2*vB^2] + 0 + Mgd

I found the answer to be:

((kd^2 - Mgd)(m+M))/m^2 = Vb

but this was not correct...please help me.