# Momentum and velocity/position vectors

1. Jul 8, 2008

### deborahlane

1. The problem statement, all variables and given/known data

I have two particles moving on a smooth horizontal table. The first, A, has mass m and the second, B, has mass 3m. A has velocity 2i + 3j and B has velocity 6i - 5j. They collide at the origin at t=0 and coalesce.

I have to 1) determine the velocity of the coalesced particle after the collision; 2) determine the amount of energy lost in the collision; 3) find expressions for rA(t) and rB(t) at time t<0; 4) find the position vector of the centre of mass of the system before the collision; 5) determine the velocity of the centre of mass; 6) find the position vector of the coalesced particles at time t>0 after the collision; and 7) comment on the answers to parts 3)-6)

2. Relevant equations

I've used the following: 1) Total linear momentum of system P=mv and found P of A is 2mi + 3mj and P of B is 18mi - 15mj, so using the Principle of conservation of momentum, I found v of coalesced particle to be 5i - 3j

2) kinetic energy=1/2mmodv2 and KE before impact is 98m and after is 68m, so loss is 30 joules.

4) rG=$$\Sigma$$miri all divided by total mass...when I finally find the position vectors!

5) Velocity of the centre of mass=total linear momentum of the system...I think.

6) With position vectors for A and B, I'm not sure what to do here: add them, since the particles coalesce?

3. The attempt at a solution

3) My problem is from 3) onwards. I'm pretty sure I can figure out the rest of it once I have 3). I can't work out how to get from the velocity vectors to position vectors. I thought of integrating and using t=0 r=0 to find a particular solution, but that seems to just leave me with rA(t)=2ti + 3tj and rB(t)=6ti - 5tj, which just doesn't seem to make sense.

Last edited: Jul 8, 2008
2. Jul 8, 2008

### Hootenanny

Staff Emeritus
Welcome to PF deborahlane and thanks for taking the time to lay out your problem properly.
Correct, just a small typo highlighted in red.
Looks good to me.

Okay for number three:
You're almost correct here. Notice that you are dealing with the case when t<0, i.e. negative time.
Again looks good.
Don't forget to divide by the total mass!
Consider the centre of mass of the system, what can you say about the velocity of the COM before and after the collision?

3. Jul 8, 2008

### deborahlane

Thanks, Hootenanny. Your suggestions have been very helpful! Sadly, though, I'm still lost on part 3)! I'll give it some more thought and see what more advice others can give, and what I can work out from your help!

4. Jul 8, 2008

### Hootenanny

Staff Emeritus
Scrap my previous comment regarding question (3), your answer is entirely correct! Sorry for the confusion.

5. Jul 8, 2008

Great! :)