How Do You Calculate Bullet Speed in a Ballistic Spring System?

[jemstone]

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

Homework Equations

KE = (1/2)mv²
U_g = mgy
U_s = (1/2)kx²

KEi + Ugi + Usi = KEf + Ugf + Usf

The Attempt at a Solution

This problem is going to end up getting really messy, but I am not sure how to incorporate the mass of the box (M) This is what I started with:

(1/2)mvi² + mgyi + (1/2)kxi² = (1/2)mvf² + mgyf + (1/2)kxf²

now I'm assuming that mgy values are 0 because we are not given any value for the height the box is above the ground. so that would give:

(1/2)mvi² + (1/2)kxi² = (1/2)mvf² + (1/2)kxf²

Incorporating the variables we are given:

(1/2)mvi² = (1/2)kd²

however this does not include gravity (which I am sure needs to be included somewhere) or the mass of the box (M)

 

[loferbri]

i don't know if this help but maybe if you put the spring at rest at y=0 and then when it's compress the y=d you could incorporate then the mgy equitation therefore putting g and M in the equation