Using the Galilean transformation and classical velocity addition

[stunner5000pt]

My problem is this:

Let's say momentum is conserved in all frames...
An observer on the ground observes two paticles with masses m1 and m2 and finds upon measurement that momentum is conserved. Use classical velocity addition to prove that momentum is conserved if the observer is on a train passing by alongside this collision.

so assume this

--------------(m2)---><------------(m1)----
<------------------Train moves in this direction
Let the initial velocity denoted as u
and final velocity as v
so if the observer is on the ground stationary

he wil see

m1u1 + m2u2 = m1v1 + m2v2

From the train
for m1 velocity is u1 - vt (where vt is velocty of train)
for m2 velocity is u2 + vt

then m1(u1-vt)+m2(u2+vt)
=m1u1 - m1vt + m2vt + m2u2 + m2vt
= m1u1 + m2u2 - m1vt + m2vt
= m1v1 + m2v2 - m1vt + m2vt

and this is where i am stuck... did i do something wrong here? or is that expression - m1vt + m2vt supposed to mean something?

Please do help

 

[TenaliRaman]

m1(v1-vt)+m2(v2+vt)

-- AI

 

[stunner5000pt]

m1(v1-vt)+m2(v2+vt)

-- AI

that's the answer isn't it?

that v1-vt represents the speed of the ball with respect to the train after it collides and it makes no different because the velocity addition applies here too, right?

 

[HallsofIvy]

Yes, that's what TenaliRaman was saying:

You arrived at
m1(u1-vt)+m2(u2+vt)= m1v1 + m2v2 - m1vt + m2vt
Now do a little rearranging on the right:
= m1v1- m1vt+ m2v2+ m2vt
= m1(v1- vt)+ m2(v2+ vt),
showing conservation of momentum.