A seemingly simple yet paradoxical question? (springs & collision)

[lillybeans]

Homework Statement

A 8g bullet hits a block of wood with mass of 450g which is at rest. The impact compresses the spring by 0.15m. The spring constant is 80N/m.

a) Find the velocity of the block and bullet when they first begin to move together.
b) Find the initial kinetic energy of the bullet.

HERE IS THE PROBLEM. I did this question before (it's from a physics test), I got it right, but now I no longer understand the reasoning behind my solution. It makes absolutely no sense.

The Attempt at a Solution

Here is what I did, and apparently, it's right. I just don't understand why.

a)Ee=1/2kx²=(1/2)(80)(0.15)²=0.9J

Ek=1/2mv²
0.9=1/2(0.458kg)v²
v=1.982m/s

b)m1v1+m2v2=m(1+2)*Vmin <--minimum separation
0.008v + 0 = 0.458*1.982
v1=113.47

Ek=1/2(0.008)(113.47)²
=51.5J

-------------------------------------

Here is why I no longer think my solution makes sense.

Total energy before collision = kinetic energy in the bullet
Total energy during collision (minimum separation)= all converted to stored energy in the spring
Total energy after collision = kinetic energy again

Problem 1: notice, during minimum separation, Et=Est, which means there IS NO kinetic energy! Which means, it's not possible to find the velocity at minimum separation since it's all stored energy and no kinetic energy. Thus, shouldn't the minimum separation velocity be zero, and not 1.982m/s?

Problem 2: If all the system's energy has been converted to stored energy in the spring during minimum separation, doesn't that mean Et=Est=0.9J? Since Et before and after collision does not change, doesn't it imply that the initial kinetic energy of the bullet is also 0.9J, and not 51.5J? Since Et=Ek initial?

 

[vela]

I know it seems perfectly obvious to you, but what exactly do E_t and E_st stand for? And what exactly is this "minimum separation" of which you speak?

 

[lillybeans]

I know it seems perfectly obvious to you, but what exactly do E_t and E_st stand for? And what exactly is this "minimum separation" of which you speak?

Yes Ma'am, My apologies!

Et= Total Energy
Est = Stored Energy
Minimum separation = The minimum distance between the two objects during collision. It is also the point where when the two objects can be thought of as a "complex" and treated as one ginormous .. ugh.. thing.

 

[vela]

I think you're just misunderstanding what happens in the problem.

The time scale of the collision is very short, so the spring doesn't compress at all during the collision. The bullet flies in and collides with the block. The block/bullet combo begins moving. It compresses the spring, which eventually brings them to rest. So immediately after the collision, there is no energy stored in the spring. It's all kinetic energy of the block/bullet combo.

By the way, I'm a dude.

 

[lillybeans]

I think you're just misunderstanding what happens in the problem.

The time scale of the collision is very short, so the spring doesn't compress at all during the collision. The bullet flies in and collides with the block. The block/bullet combo begins moving. It compresses the spring, which eventually brings them to rest. So immediately after the collision, there is no energy stored in the spring. It's all kinetic energy of the block/bullet combo.

By the way, I'm a dude.

Thank you for answering, but what does it mean when the question asks "find the velocity when the block and the bullet first begin moving together"? Is that when the bullet has JUST touched the block right before it compresses the spring? Or is it AFTER the spring has been compressed, right before the two are about to launch off/separate again?

Also, another question. Assuming part b)'s answer is correct, that means we started off with 51.5J of kinetic energy in the bullet, and during the spring compression phase, where all the kinetic energy is converted to stored energy, we only have 0.9J of energy in total? Where did the 50.6J of energy go? And I'm pretty sure this is an elastic collision (no energy is lost)?

P.S. I'm sorry. I thought Vela is a woman's name.

 

[gneill]

I believe that the implication is that the bullet becomes embedded in the block of wood. Thus the situation is that of an inelastic collision. Energy will be lost.