What is the ratio of masses for two colliding gliders in a horizontal air flow?

[pablotano]

Homework Statement

Two gliders are free to move in a horizontal air through. One is stationary and the other one collides perfectly ellastically. They rebound with equal and opposite velocities. What is the radio of their masses?

The Attempt at a Solution

The answer is 3, how can I solve the problem?

 

[Nathanael]

Do you know about conservation of momentum?

What does it mean to collide perfectly elastically?

 

[pablotano]

I understood that momentum is conserved in perfectly ellastically collisions. In vol 1 of the feynman lectures, chapter 10, he says that if two equally massed particles collide, they simple change velocities between them. This problem would be easy for me if the gliders would “collide and stick“ rather than bounce. The “bouncing“ behavior is the problem for me here..

 

[Nathanael]

I understood that momentum is conserved in perfectly ellastically collisions.

Momentum is conserved in ALL collisions.

Perfectly elastic collisions are just a special kind of collision. What is unique about perfectly elastic collisions?

 

[pablotano]

Kinetic energy is conserved in this kind of collisions, I've already tried to solve the problem equalling the kintetic energy of the system before and after the collisions, but I cannot see the relationship between the inicial velocity of the particle I think is the lighter one with the final velocity of the two particles.

 

[Nathanael]

Do you mind showing your math?

 

[pablotano]

I'm sorry Nathanael, but I can't. I've been around this problem for more than an hour now (my physics education is just getting started), I can't write the math because I do not understand what happen with the velocities after this kind of collision (when the collision is between different masses)

 

[Nathanael]

Do you know how to write "momentum is conserved" and "kinetic energy is conserved" in math?

Use $V_i$ for the initial velocity of the one glider and $V_f$ for the final velocity of both gilders.

There are two unknowns ($V_i$ and $V_f$) and two restrictions (conservation of KE and momentum) so you should be able to solve the problem

 

[pablotano]

Ok, here is what I did:

Conservation of KE (initial velocity is vo, final velocity is vf and R is the ratio of the masses):

(m*(vo)^2)/2 = (Rm*vf^2)/2 + (m*vf^2)/2
...
R=(vf^2)/(vo^2-vf^2)

Since I do not know the relationship between vo and vf, I didn't get to the answer.

 

[Nathanael]

I think your second equation is wrong:

R=(vf^2)/(vo^2-vf^2)

Try using "m" and "M" instead of "R"
(and then, if you wish, you can replace "M" with "mR")

Since I do not know the relationship between vo and vf, I didn't get to the answer.

Now you don't know the relationship between $V_i$ and $V_f$ but the relationship is there in the mathematics.
To use it, just solve for $V_i$ (or $V_f$) and plug it into the other equation.

 

[pablotano]

Thanks! Everithing is starting to look better.
For the conservation of momentum, it would be:

m*Vo=M*Vf + m*Vf

?

or

m*Vo=M*Vf - m*Vf

 

[Nathanael]

Thanks! Everithing is starting to look better.
For the conservation of momentum, it would be:

m*Vo=M*Vf + m*Vf

?

Almost! But you forgot about the direction. The velocities Vf (of M and m) have the same magnitude, but they have the opposite direction.EDIT:
Sorry I didn't see this:

or

m*Vo=M*Vf - m*Vf

Yes, that would be the correct equation. (You could factor it a bit giving you $mV_o=V_f(M-m)$)Now try plugging in Vo or Vf into the other equation and see if you can get to the answer.

 

[pablotano]

Thanks for teaching me! :) I finally did it.

Now as a commentary, I was punishing my brain with velocities relationships because in Feynman lectures, he solves every "collision" problem using 'Galilean Relativity'. That is why I thought that I wasn't seeing something, I tried to think of this problem as if I was seeing it from a moving car, etc.

I find this way of solving the problem easier, it is even possible to solve this using just 'Galilean Relativity', by comparing collisions as seen from a moving car with collisions seen from the 'centers of mass' ?

 

[Nathanael]

I'm sure there is a way to approach the problem intuitively, but I haven't been able to think of a good way.

The way I showed you to solve the problem is the "last chance" "nothing else is working" kind of method.

I'll be honest, the only reason I showed you how to solve it in this way is because all of my attempts to simplify the problem into something more intuitive have failed.

(Failed in the sense of making the math simpler. Thinking about it has given me a bit of an understanding of what's going on, but not to the extent that I can explain why it is reasonable for an object colliding elastically into an object 3 times the mass results in equal speeds)


P.S.
Where did you get problems for the feynman lectures? I've found the lectures but not any problem sets.