Which Collision Results in Greater Momentum and Energy Loss?

In summary: This is not a trick question. But teachers are fond of assigning "trick" questions.)In summary, the problem involves two cases where two carts with identical mass collide. In the first case, the carts stick together after one cart starts at a height above the ground and crashes into the stationary cart on flat ground. In the second case, the carts start at the same height on opposite sides and fall in opposite directions before colliding and sticking together. The question asks which case has a greater magnitude of total momentum and which case has a greater loss of mechanical energy. The total momentum in case 2 is zero due to the opposite directions, while the loss of mechanical energy is greater in case 2 because all of the mechanical energy is
  • #1
justagirl
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Say there was 2 collisions. There are 2 carts of identical mass. In the first case cart A starts at height h above the ground and crashes into cart B which is stationary on the flat ground. They stick together.

In the second case, cart A starts at height h/2 above the ground, cart B starts at height h/2 above on the ground on the other side. They fall in opposite directions and collide and stick together.

The problem asks that when the carts reach the flat ground, in which case is the magnitude of the total momentum greater.

When they ask about the magnitude, in case 2, should the momentum be
M(a)V(a) + M(b)V(b) = 2M(a)V(a) (which would be greater than case 1), or should I factor in the fact that they are traveling in opposite directions, in which case the total momentum would be zero?

The second question asks whether the total energy lost in greater in case 1 or case 2. The energy lost would be greater in case 2 if the momentum was 2M(a)V(a)... but then again this goes to question 1.

Thanks!
 
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  • #2
justagirl said:
When they ask about the magnitude, in case 2, should the momentum be
M(a)V(a) + M(b)V(b) = 2M(a)V(a) (which would be greater than case 1), or should I factor in the fact that they are traveling in opposite directions, in which case the total momentum would be zero?

At the time of writing, this is a 17+ year old question. So I'm giving a fairly complete answer/explanation.

You must account for the opposite directions. The total momentum for case-2 is therefore zero. That's because momenta are vectors; they must be added as vectors. (If the magnitude of the resultant is needed, this is found after the vector-addition.)

That means forcase-2, the carts (now stuck together) will end-up stationary. You can reach the same conclusion by thinking about symmetry. There is no preferred direction of motion for the combined carts in case-2.

justagirl said:
The second question asks whether the total energy lost in greater in case 1 or case 2. The energy lost would be greater in case 2 if the momentum was 2M(a)V(a)... but then again this goes to question 1.
This is badly worded (or maybe a trick) question. Energy is always conserved. So the total energy loss in both cases is zero.

However, the question was probably intended to be about loss of mechanical energy (ME). Mechanical energy is potential energy + kinetic energy.

In both cases, the starting ME is the same:
- for case-1 it is mgh+ 0 = mgh
- for case-2 it is mg(h/2) + mg(h/2) = mgh

In case-2 all the ME is lost, as the carts end-up stationary. (The ME gets entirely converted to heat (and maybe a little sound energy) as a result of the collision.)

In case 1, the stuck-together cars are still moving. So not all of the ME has been converted to heat.

So the loss of ME is greater in case 2.
 
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Related to Which Collision Results in Greater Momentum and Energy Loss?

What is total momentum?

Total momentum is the product of an object's mass and its velocity. It is a vector quantity that describes the amount of motion an object has in a specific direction.

How is total momentum calculated?

Total momentum is calculated by multiplying an object's mass (m) by its velocity (v). The equation for calculating total momentum is p = m * v.

What is the relationship between total momentum and mass?

The total momentum of an object is directly proportional to its mass. This means that as an object's mass increases, its total momentum also increases.

What is the relationship between total momentum and velocity?

The total momentum of an object is directly proportional to its velocity. This means that as an object's velocity increases, its total momentum also increases.

Why is the magnitude of total momentum important?

The magnitude of total momentum is important because it helps to quantify the motion of an object. It is a useful concept in understanding and predicting the behavior of objects in motion, such as in collisions or when calculating the forces acting on an object.

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