Elastic collision heavy particle problem

[Abu]

Homework Statement

Suppose a heavy particle (mass m1) has an elastic head-on collision with a very light particle of mass m2 initially at rest. Show that if m1>>m2, the velocity of the projectile (m1) is practically unchanged, whereas the target particle (m2) acquires a velocity v2' = 2v1

Homework Equations

m1v1+ m2v2 = m1v1'+m2v2'
v1-v2 = v2'-v1'

The Attempt at a Solution

When I first tried to solve it I thought I had it correct.
Basically I did something really simple like this:
v1-v2 = v2'-v1'
v1 - 0 = v2'-v1'
v1 = v2'-v1'

Then here I just went to rearrange v1' with v1, forgetting that they are not the same values. Then I got the 2v1 = v2' but I did it incorrectly, obviously. I don't know where to go from here though... Thank you.

 

[DoItForYourself]

What about the kinetic energy of the system? What do you know about the kinetic energy in elastic collisions?

 

[Abu]

What about the kinetic energy of the system? What do you know about the kinetic energy in elastic collisions?

Oh right... kinetic energy in elastic collisions is conserved. I know the equation of it, if I type it though it will look really messy. Here it is though:

1/2m1v1^2 + 1/2m2v2^2 = 1/2m1v1'^2 + 1/2m2v2'^2

 

[DoItForYourself]

Hint 1 : Initially use the momentum conservation principle to show that the velocity of 1 is practically unchanged.

Hint 2 : Take v₁^' from the first equation and put it in the kinetic energy conservation equation.

 

[Abu]

Hint 1 : Initially use the momentum conservation principle to show that the velocity of 1 is practically unchanged.

Hint 2 : Take v₁^' from the first equation and put it in the kinetic energy conservation equation.

Okay so what I did was solve for the velocity of the large mass after the collision. What I found was that the velocity of the large mass after the collision is it's velocity before the collision minus the momentum of the small mass after the collision. It looked like this
m1v1 +m2v2 = m1v1' + m2v2'
m1v1 + 0 = m1v1' + m2v2'
v1' = v1 - m2v2'

Does this prove the first hint? Because won't the momentum of the smaller mass after the collision still be a lot smaller than the speed of the larger mass before the collision? I feel like I didn't get it right.

 

[DoItForYourself]

You have an error just before you form your final equation. Check it again.

 

[ehild]

Homework Statement

Suppose a heavy particle (mass m1) has an elastic head-on collision with a very light particle of mass m2 initially at rest. Show that if m1>>m2, the velocity of the projectile (m1) is practically unchanged, whereas the target particle (m2) acquires a velocity v2' = 2v1

Homework Equations

m1v1+ m2v2 = m1v1'+m2v2'
v1-v2 = v2'-v1'

The Attempt at a Solution

When I first tried to solve it I thought I had it correct.
Basically I did something really simple like this:
v1-v2 = v2'-v1'
v1 - 0 = v2'-v1'
v1 = v2'-v1'

Then here I just went to rearrange v1' with v1, forgetting that they are not the same values. Then I got the 2v1 = v2' but I did it incorrectly, obviously. I don't know where to go from here though... Thank you.

You can not simply replace v1' with v1. Use also the first equation m1v1+ m2v2 = m1v1'+m2v2' --->m1v1=m1v1'+m2v2' to get v1' and v2'.
After that, you can see what happens when m1>>m2.

 

[ehild]

What about the kinetic energy of the system? What do you know about the kinetic energy in elastic collisions?

Abu got the second equation in the OP by using conservation both of momentum and kinetic energy.

 

[Abu]

You can not simply replace v1' with v1. Use also the first equation m1v1+ m2v2 = m1v1'+m2v2' --->m1v1=m1v1'+m2v2' to get v1' and v2'.
After that, you can see what happens when m1>>m2.

I'm getting confused, I'm really sorry.

So i did what you said and I got
v1' = m1v1 - m2v2'/m1
and
v2' = m1v1 - m1v1'/m2

But I don't see how that can tell me what happens when m1>>m2.

 

[DoItForYourself]

Abu got the second equation in the OP by using conservation both of momentum and kinetic energy.

Oh yes, you are right!

But still, I think he must show how he reached that point (that he started with the kinetic energy conservation law).

 

[DoItForYourself]

Abu,

please check your equations again. You did an error when dividing by m₁ and m₂.

 

[ehild]

I'm getting confused, I'm really sorry.

So i did what you said and I got
v1' = m1v1 - m2v2'/m1
and
v2' = m1v1 - m1v1'/m2

But I don't see how that can tell me what happens when m1>>m2.

Use the other equation, too, that you got in the OP : v1 = v2'-v1', together with the equation m1v1 = m1v1'+m2v2'.
v1 is given. Write v1' and v2' in terms of v1 and the masses.

 

[ehild]

Oh yes, you are right!

But still, I think he must show how he reached that point (that he started with the kinetic energy conservation law).

You are right, but certainly he was taught how to get that equation in case of elastic collision.