Question

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A spring whose spring constant k is 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.

Answer 1

to begin with the spring was at rest even with the weight mg acting down
so immediately there is a built-in restoring force of mg in the spring

kinetic energy of the bullet is used to do work against gravity(-mgd) which is stored as +mgd
and also to do work by spring force -kx
=1/2kx^2
so kinetic====w.d by gravity+w.d by spring force

Answer 2

bullet-block
m*v=(M+m)*V =>V=.....

spring-block(no bullet yet)
M*g=k*d1 => d1=......

systemw energy after bullet
Einitial=Efinal
1/2*(M+m)*V^2 + 1/2*k*d1^2= (M+m)*g*d + 1/2*k*(d1-d)^2