Momentum and velocity/position vectors

[deborahlane]

Homework Statement

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 r_A(t) and r_B(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)

Homework 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) r_G=\Sigmam_ir_i 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?

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 r_A(t)=2ti + 3tj and r_B(t)=6ti - 5tj, which just doesn't seem to make sense.

 

[Hootenanny]

Welcome to PF deborahlane and thanks for taking the time to lay out your problem properly.

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

Correct, just a small typo highlighted in red.

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

Looks good to me.

Okay for number three:

3) '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 r_A(t)=2ti + 3tj and r_B(t)=6ti - 5tj, which just doesn't seem to make sense.

You're almost correct here. Notice that you are dealing with the case when t<0, i.e. negative time.

4) r_G=\Sigmam_ir_i all divided by total mass...when I finally find the position vectors!

Again looks good.

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

Don't forget to divide by the total mass!

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

Consider the centre of mass of the system, what can you say about the velocity of the COM before and after the collision?

 

[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!

 

[Hootenanny]

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!

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

 

[deborahlane]

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

Great! :)