M4 Elastic Collisions

Could someone please help me with this question?

A smooth ball of mass 0.5kg moves in the xy plane. The ball collides with a fixed vertical wall containing the line x + y = 3. Velocity of ball before impact = (-4i - 2j) metres per second. Coefficient of restitution between the ball and the wall = 0.5.

I need to find the velocity of the ball after the impact. I believe I need to resolve parallel and perpendicular to the wall, but I am stuck on how to do this.

Reply 1

ghostwalker

Original post by EXTREMEninja

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How do you want to do it - in terms of vectors or coordinate geometry; you have a mix of both there?

Reply 2

EXTREMEninja

Original post by ghostwalker

How do you want to do it - in terms of vectors or coordinate geometry; you have a mix of both there?

I do not mind, perhaps you could explain both ways, if you have time? My textbook does not really provide any examples of this type of problem :/

Reply 3

ghostwalker

Original post by EXTREMEninja

I do not mind, perhaps you could explain both ways, if you have time? My textbook does not really provide any examples of this type of problem :/

I'll look at a vector method:

In either case you will find a diagram helpful.

Consider:

What's a direction vector for the wall?
Then using the dot product you can work out the component of the velocity parallel to the wall.
Subtract that from your original velocity to get the component perpendicular to the wall.
After impact the component perpendicular to the wall is ...?
Add in the parallel component and bingo.

Let me know if you need me to expand on any of those bits - but have a think first.

Reply 4

EXTREMEninja

Original post by ghostwalker

Ah yes, should I work out a unit vector parallel to the wall? How would I use the dot product to work out the component? (I know what the dot product is, just not how to use it here).

Reply 5

ghostwalker

Original post by EXTREMEninja

Ah yes, should I work out a unit vector parallel to the wall? How would I use the dot product to work out the component? (I know what the dot product is, just not how to use it here).

Yes a unit vector is easier to work with.

If e is your unit vector, then the component parallel to the wall is (v.e)e.

v.e being the scalar multiple of the vector e.

Reply 6

EXTREMEninja

Original post by ghostwalker

Yes a unit vector is easier to work with.

If e is your unit vector, then the component parallel to the wall is (v.e)e.

v.e being the scalar multiple of the vector e.

Oh, I think I see now. Can I also apply the same method to a unit vector perpendicular to the wall (for the perpendicular component) or do I then need to take into account the coefficient of restitution?

Reply 7

ghostwalker

Original post by EXTREMEninja

You could use a unit vector perpendicular to the wall, dot product to find the component of the velocity perp. to the wall, etc.

Though, it's not necessary, as the sum of the parallel and perp. components is equal to the original velocity (as vectors). So once you know one, you can do a subtraction to get the other.

After impact the component perp to the wall will be -e times the component perp. to the wall before impact.

Reply 8

EXTREMEninja

Original post by ghostwalker

Okay, so far I have this. Unit vector parallel to wall = (i - j) = e.

e.v = -sqrt(2). I then multiply this by the unit vector, and then subtract from the original velocity?

Note: Forgot the 1/sqrt(2) in the unit vector.

(edited 9 years ago)

Reply 9

ghostwalker

Original post by EXTREMEninja

Okay, so far I have this. Unit vector parallel to wall = (i - j) = e.

e.v = -sqrt(2). I then multiply this by the unit vector

Yes, that's the scalar multiple of the unit vector.

, and then subtract from the original velocity?

Note: Forgot the 1/sqrt(2) in the unit vector.

Subtacting that from the original vector gives you the component perp. to the wall, yes.

Reply 10

EXTREMEninja

Original post by ghostwalker

Yes, that's the scalar multiple of the unit vector.

Subtacting that from the original vector gives you the component perp. to the wall, yes.

So I end up with, (-3i - 3j) after doing (-4i - 2j) - parallel component. Then I use the restitution stuff to get the perpendicular component after impact?

So, perpendicular velocity after impact = 1/2(-3i - 3j)?

Reply 11

ghostwalker

Original post by EXTREMEninja

You missed the minus sign on that last bit - you multiply by "-e"

Edit: Last one for to-night.

Reply 12

PhysGeek

I'm sorry if this is very unhelpful and slightly annoying but: I really want to do M3 and M4 as part of my fm course but my year isn't doing any more mechanics. How difficult do you think it would be to teach myself?? Thank you, and sorry to bother you. Xx

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Reply 13

EXTREMEninja

Original post by ghostwalker

You missed the minus sign on that last bit - you multiply by "-e"

Edit: Last one for to-night.

Okay, why is it -e? Thank you very much for your help.

Reply 14

ghostwalker

Original post by EXTREMEninja

Okay, why is it -e? Thank you very much for your help.

OK, last, last one for today.

The component of velocity perpendicular to the wall, as well as being multiplied by e, also reverses direction (which is where the minus sign comes in). E.g. A ball hitting a wall - it bounces back.