Comparing Collisions: Momentum and Energy Conservation

[dherm56]

Homework Statement

Cart A and B are identical. Consider the two collisions described below.

In Collision 1, Cart A starts from rest on a hill at height h above the ground and then collides with Cart B which is initially at rest on the ground. The two carts stick together.

In Collision 2, Carts A and B are at rest on opposite hills at heights h/2 above the ground. They roll down, collide head-on with each other on the ground and stick together.

1) Once the carts have reached the ground the magnitude of the total momentum of the two-cart system in Collision 2 is:

less than the magnitude of the total momentum of the two-cart system in Collision 1.
equal to the magnitude of the total momentum of the two-cart system in Collision 1.
greater than the magnitude of the total momentum of the two-cart system in Collision 1.

2) The energy lost in Collision 2 is greater than the energy lost in Collision 1.

true
false

Homework Equations

Conservation of momentum. P=mv

The Attempt at a Solution

For 1, the momentum has to be conserved, so I chose B the two momentums are equal in magnitude.

For 2, I am less certain. I chose True because collision two is enalstic and I assume there would be more energy lost to sound in the collision.

Any suggestions?

 

[Doc Al]

For 1, the momentum has to be conserved, so I chose B the two momentums are equal in magnitude.

In each collision, momentum is conserved. But that doesn't mean the momentum in those two different situations is equal. Try and figure it out.

For 2, I am less certain. I chose True because collision two is enalstic and I assume there would be more energy lost to sound in the collision.

Both collisions are inelastic, so there's more to it than that.

 

[Queue]

Homework Statement

Cart A and B are identical. Consider the two collisions described below.

In Collision 1, Cart A starts from rest on a hill at height h above the ground and then collides with Cart B which is initially at rest on the ground. The two carts stick together.

In Collision 2, Carts A and B are at rest on opposite hills at heights h/2 above the ground. They roll down, collide head-on with each other on the ground and stick together.

1) Once the carts have reached the ground the magnitude of the total momentum of the two-cart system in Collision 2 is:

less than the magnitude of the total momentum of the two-cart system in Collision 1.
equal to the magnitude of the total momentum of the two-cart system in Collision 1.
greater than the magnitude of the total momentum of the two-cart system in Collision 1.

2) The energy lost in Collision 2 is greater than the energy lost in Collision 1.

true
false

Homework Equations

Conservation of momentum. P=mv

The Attempt at a Solution

For 1, the momentum has to be conserved, so I chose B the two momentums are equal in magnitude.

For 2, I am less certain. I chose True because collision two is enalstic and I assume there would be more energy lost to sound in the collision.

Any suggestions?

You forgot a couple of equations. Gravitational potential energy and kinetic energy to be exact (mgh and .5mv²).

What is the velocity of cart 1 just before the crash in the first collision? What is the velocity of one of the two carts before collision in the second collision?

 

[Doc Al]

Here's a hint for case 2: How do the speeds of carts A and B compare just before they collide? What's their total momentum?

(You don't need to do any calculation to solve this problem. Just a bit of clear thinking.)

 

[dherm56]

for case 1, the second collision has a smaller net magnitude because the system is not moving where as the first collision the carts continue in a positive direction. Therefore making the answer A

for case 2, the speed of the cars are exactly the same and therefore stop when they collide. Because in collision 2 the carts stop and in collision 1 the carts continue moving more energy is lost in collision 2.

Am I thinking correctly?

 

[Doc Al]

Excellent.

One step in the reasoning that you didn't make explicit: How does the total mechanical energy of the two situations compare?

 

[dherm56]

mechanical energy I am a little lost on. What is the discrepancy between that and momentum? I always thought total mechanical energy = total momentum

 

[Doc Al]

I always thought total mechanical energy = total momentum

No. Energy and momentum are two different things. For one thing, energy is a scalar (direction doesn't matter) while momentum is a vector. The two cars moving in opposite directions will have zero total momentum but plenty of mechanical energy. (Until they crash, and that mechanical energy is transformed into heat and deformation.)

In this problem, the thing to realize is that both cases start out with the exact same amount of mechanical energy:
Case 1 = mgh
Case 2 = mgh/2 + mgh/2 = mgh = same as Case 1.

 

[dherm56]

So ultimately even though momentum is different, total mechanical energy is the same before and after the collision?

 

[Doc Al]

So ultimately even though momentum is different, total mechanical energy is the same before and after the collision?

No! While momentum is conserved in any collision, mechanical energy is only conserved in elastic collisions. The collisions in this problem are perfectly inelastic (they stick together). So some (or all) of the mechanical energy is "lost" in these collisions.

 

[dherm56]

Ooooh!
Alright, that clears a lot of things up.
Thank you very much!