Finding Velocity of a bullet using springs and hanging masses

[viciousp]

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.

Homework Equations

KEi + PEi = KEf +PEf
F= -kx

The Attempt at a Solution

I first used Newtons second law to find the the distance x the string is stretched, so:
Mg=k(x-d)
x= (Mg/k)+d

Then I used the energy law like so:
1/2(m_b(V_b)² +Mgy + 1/2(k)(x²) = Mg(y+d)

solving for V_b I get:

sqrt[(2Mgd-(k(d+Mg/k)^2))/m]

But I am pretty sure this is wrong, can someone help me

 

[Delphi51]

I don't fully understand this
1/2(mb(Vb)2 +Mgy + 1/2(k)(x2) = Mg(y+d)
The distance y is not clear to me.
It seems to me that it should be
energy before = energy after
½mv² + ½kx² = Mgy + ½k(y-x)²
assuming that y is the amount M moves upward when the bullet hits

 

[viciousp]

I used y as the height from the block to the ground

 

[Delphi51]

Okay, that makes sense but your formula does not include the energy stored in the spring after the collision.

Actually, I'm worried about the whole approach. The bullet will embed itself in the block, losing lots of energy to friction. We have no way to estimate this loss of energy so we can't use conservation of energy.

Recommend you use conservation of momentum in the collision of the bullet with the block. That should give you the speed and kinetic energy the bullet/block after the collision.