Conservation of momentum/kinetic energy in inelastic collisions.

In summary: Thanks a lot.In summary, the conversation discusses the concept of conservation of momentum and kinetic energy in inelastic collisions. While momentum is conserved due to the center of mass, kinetic energy is not due to the loss of energy in the form of sound and heat. The velocity of the center of mass remains constant, but the total kinetic energy of the system is not equal to 1/2 (m * V^2). This is because averages do not commute in general.
  • #1
Sweeney
5
0
I'm finding it difficult to understand how momentum is conserved in an inelastic collision, but kinetic energy isn't. I understand some energy leaves the system in the form of sound and heat.

If we look at the momentum of the system before and after the collision, P=mV, they are equal, due to CoM. Assuming a constant mass we can also say that the velocity of the system is constant.

Now looking at kinetic energy, E=(1/2)mv^2 we know that it changes. Again assuming a constant mass we can say that v^2 changes and thus v changes. Obviously these two statements contradict each other, but I don't understand where I went wrong.
 
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  • #2
I'll try and think of a good way to put it... but in response to your first line, inelastic collisions are defined by a non-conservation of kinetic energy. That being said, is your question really, "why is kinetic energy conserved in some, but not all, collisions?"
 
  • #3
Sweeney said:
If we look at the momentum of the system before and after the collision, P=mV, they are equal, due to CoM. Assuming a constant mass we can also say that the velocity of the system is constant.
The velocity of the system's center of mass remains constant.

Now looking at kinetic energy, E=(1/2)mv^2 we know that it changes. Again assuming a constant mass we can say that v^2 changes and thus v changes. Obviously these two statements contradict each other, but I don't understand where I went wrong.
The total kinetic energy of the system = Ʃ(1/2)mv^2 for each mass element. v is not the velocity of the center of mass.
 
  • #4
Sweeney said:
I'm finding it difficult to understand how momentum is conserved in an inelastic collision, but kinetic energy isn't. I understand some energy leaves the system in the form of sound and heat.

If we look at the momentum of the system before and after the collision, P=mV, they are equal, due to CoM. Assuming a constant mass we can also say that the velocity of the system is constant.

If there is a collision then there are at least three quantities that can be meant by "m" and at least three quantities that can be meant by "V".

It would seem that you are taking m to denote the sum of the masses of the colliding objects and V to denote the velocity of the center-of-mass of the colliding objects.

For momentum, that works out because P(total) = m(total) * V(average)

For energy, this does not work because E(total) != 1/2 m(total) * V(average)^2

Let's take an more familiar example... Suppose that you have a pile of building blocks. They are all shaped like cubes. Some are big and some are little. If you pile all the blocks on top of one another you can determine the height of the tower by multiplying the average size of each block by the number of blocks. But you cannot correctly determine the total volume of the blocks by taking the average size the blocks, cubing that and multiplying by the number of blocks.

The average of the cubes is not equal to the cube of the average.
The average of the squares is not equal to the square of the average.
The average of the inverse is not equal to the inverse of the average.

Averages don't commute in general.

That's why there are things like the "harmonic mean" or the "root mean square".
 
  • #5
Absolutely perfect answer! I finally can understand it.
 

1. What is the conservation of momentum/kinetic energy in inelastic collisions?

The conservation of momentum/kinetic energy in inelastic collisions is a fundamental principle in physics that states that the total momentum and kinetic energy of a closed system remains constant before and after the collision.

2. How does the conservation of momentum/kinetic energy apply to inelastic collisions?

In an inelastic collision, objects stick together or deform upon impact, causing a loss of kinetic energy. However, the total momentum of the system remains unchanged, according to the principle of conservation of momentum. This means that the combined momentum of the objects before the collision is equal to the combined momentum after the collision.

3. What is the formula for calculating the conservation of momentum/kinetic energy in inelastic collisions?

The formula for calculating the conservation of momentum in inelastic collisions is: m1v1 + m2v2 = (m1 + m2)v, where m1 and m2 are the masses of the objects, v1 and v2 are their velocities before the collision, and v is their combined velocity after the collision. The formula for calculating the conservation of kinetic energy is: 1/2m1v1^2 + 1/2m2v2^2 = 1/2(m1 + m2)v^2.

4. What factors can affect the conservation of momentum/kinetic energy in inelastic collisions?

The conservation of momentum/kinetic energy in inelastic collisions can be affected by factors such as the masses and velocities of the objects involved, as well as external forces acting on the system, such as friction or air resistance. In addition, the type of collision (head-on, glancing, etc.) and the materials of the objects can also impact the conservation of momentum/kinetic energy.

5. Can the conservation of momentum/kinetic energy be violated in inelastic collisions?

No, the conservation of momentum/kinetic energy is a fundamental law in physics and cannot be violated. In any inelastic collision, the total momentum and kinetic energy of the system will remain constant, even if there is a loss of energy due to deformation or other external factors.

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