M4 Elastic Collisions

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EXTREMEninja
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#1
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#1
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.
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ghostwalker
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#2
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#2
(Original post by EXTREMEninja)
...
How do you want to do it - in terms of vectors or coordinate geometry; you have a mix of both there?
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EXTREMEninja
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#3
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#3
(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 :/
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ghostwalker
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#4
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(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.
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EXTREMEninja
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#5
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#5
(Original post by ghostwalker)
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.
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).
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ghostwalker
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#6
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#6
(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.
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EXTREMEninja
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#7
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#7
(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?
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ghostwalker
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#8
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#8
(Original post by EXTREMEninja)
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?
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.
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EXTREMEninja
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#9
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#9
(Original post by ghostwalker)
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.
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.
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ghostwalker
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#10
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#10
(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.
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EXTREMEninja
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#11
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#11
(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)?
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ghostwalker
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#12
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#12
(Original post by EXTREMEninja)
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)?
You missed the minus sign on that last bit - you multiply by "-e"

Edit: Last one for to-night.
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PhysGeek
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#13
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#13
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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EXTREMEninja
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#14
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#14
(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.
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ghostwalker
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#15
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#15
(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.
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EXTREMEninja
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#16
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#16
(Original post by ghostwalker)
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.
Ah I see. Thanks for clearing that up
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ghostwalker
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#17
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#17
(Original post by 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
Can't comment myseif, but:

Best create a new thread with an appropriate title - your enquiry will get lost buried in this thread.
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