# Conservation momentum using center of mass

## Homework Statement

A very massive object with a velocity "v" collides head-on with an object at rest whose mass is very small. no kinetic energy is converted into other forms. Prove that the low-mass object recoils with a velocity 2"v"
use the center-of-mass frame of reference.

p=mv
ke= 1/2mv^2
M1>m2

## The Attempt at a Solution

my book doesn't actually describe the center of mass frame of reference so I'm lost there. But trying to solve for it:

m1*v1=m1*v2+m2*2*v1
m1(v1-v2)= m2*2*v1
(v1-v2)m1/m2=2*v1

conservation of momentum
m1*v1 + m2*0= m1*v2+ m2*2*v1
m1*v1 = m1*v2+ m2*2*v1
m1*v1 - m2*2*v1 = m1*v2
v1(m1 - m2*2)/m1 = v2

conservation of energy
1/2*m1*(v1)^2 +1/2*m*0= 1/2*m1*(v2)^2 +1/2*m2*(2v1)^2
1/2*m1*(v1)^2 = 1/2*m1*(v2)^2 +1/2*m2*(2v1)^2
1/2*m1*(v1)^2 = 1/2*m1*(v1(m1 - m2*2)/m1)^2 +1/2*m2*(2v1)^2
m1*(v1)^2 = m1*(v1(m1 - m2*2)/m1)^2 +m2*(2v1)^2
0 = m1*(v1(m1 - m2*2)/m1)^2 +m2*(2v1)^2 - m1*(v1)^2

gneill
Mentor
The words, "A very massive object" is code for: "it's so massive that nothing that happens in the problem will measurably affect its momentum".

In this case the initially moving object, m1, is that very massive object. As such it is essentially the center of mass of the objects comprising the system. So if you, the observer, were to take up a position sitting on that massive object and you constructed your coordinate axes there, your point of view would be from the center of mass frame of reference.

From that point of view the very massive object m1 is at rest. What's the initial velocity of m2 in that frame of reference?

Why couldn't my book have anything like that explanation.....

M2 in this case would have the same velocity as the super massive object (m1) so -v1, so when it collides it becomes v1? I don't get the 2*v1, unless we are talking about the total change in vector. Which would make sense 1- to 1 is 2

gneill
Mentor
Why couldn't my book have anything like that explanation.....

M2 in this case would have the same velocity as the super massive object (m1) so -v1, so when it collides it becomes v1? I don't get the 2*v1, unless we are talking about the total change in vector. Which would make sense 1- to 1 is 2

You're almost there. You've determined that the velocity of m2, after collision, will be vV1 in the center-of-mass frame of reference; After all, it's just behaving like a ball bouncing off of a wall.

Now comes the "magic" that is the usefulness of the center of mass frame. If you now switch back to the original frame of reference mass m1 is observed to be still moving at velocity v1 (it is, after all, a "very massive object"!), but what velocity will m2 have? Remember, it's moving at speed v1 with respect to m1.

the velocity of M2 is now V1 for the center of mass reference point. But if we let that continue to go at v1 then the two vectors add V1+V1 = 2 *V1, or V2

:D

Woot!!!! Thanks for the help :D