Why Are Relative Velocities in Elastic Collisions Frame-Independent?

[gibberingmouther]

Homework Statement

"For a two-particle interaction, the relative velocity between the two vectors is independent of the choice of relatively inertial reference frames."
and
"The change in kinetic energy is independent of the choice of relatively inertial reference frames."

My textbook says that if you look at the velocities of a two particle elastic interaction before and after the collision, regardless of which reference frame you choose, the relative velocities before and after will be the same. It just says this, and I guess expects you to make an intuitive leap? I wanted a more mathematical explanation which is why i looked on the internet and found the MIT page. The professor gives an example in the case of a two particle interaction where the one particle is twice the mass of the other, and shows that in this case the relative velocities will be the same. Cool, but i wasn't really satisfied.

Anyway, besides hoping for a better explanation of the relative velocities in an elastic collision, I was also confused by the terminology of "a relatively inertial reference frame". I want to read and understand the MIT pdf but i need to be able to know what that terminology means first. I am also hoping someone could help me by explaining the two statements from the pdf i quoted above.

http://web.mit.edu/8.01t/www/materials/modules/chapter15.pdf

Homework Equations

conservation of kinetic energy and conservation of momentum.

The Attempt at a Solution

i tried to imagine two coordinate systems moving relative to each other and the velocities being different while the relative velocities are the same. couldn't quite manage it. maybe i should drink more coffee, lol? I'm grateful for my intelligence but I'm not quite an Einstein :(

i was able to read some of the math in the MIT pdf but still trudging through, just need to get unstuck on the parts i mentioned above.

 

[haruspex]

two coordinate systems moving relative to each other and the velocities being different while the relative velocities are the same.

If in the first frame the velocities are $\vec u_1$ for particle 1 and $\vec u_2$ for particle 2, what is the velocity of particle 2 relative to particle 1 in that frame?
If a second frame has velocity $\vec f$ relative to the first frame, what is the velocity of particle 1 in that frame?
Likewise, particle 2 in the second frame?

relatively inertial reference frame"

I believe that means the frames are not accelerating relative to each other. So they need not be inertial franes, but whatever acceleration they have is the same for each.

 

[Orodruin]

The easiest way of seeing that the relative velocity must remain the same after the collision is to go to the centre of mass frame where the total momentum is zero. This means that both objects must have momenta that are equal in magnitude but opposite in direction.

 

[PeroK]

"The change in kinetic energy is independent of the choice of relatively inertial reference frames."

It looks like you have actually several questions here.

1) It might be a good exercise to show that if momentum is conserved in one inertial frame of reference, then it is conserved in all inertial frames of reference.

Hint: use the idea @haruspex gave you above for 3D motion.

2) Then, you might like to show that if momentum is conserved, then the change in kinetic energy is the same in all inertial reference frames.

Hint: try the case of 1D motion first and then extrapolate to 3D.

The above is true for any interaction of two or more particles.

3) A collision of two particles is elastic if kinetic energy is conserved, which is equivalent to the separation speed between the particles being the same before and after the collision.

Again, it's a good exercise to show this. Hint: see @Orodruin 's post.

 

[neilparker62]

I wanted a more mathematical explanation which is why i looked on the internet and found the MIT page.

The maths is there on page 15_6 of your very excellent reference.