Math Is Fun Forum

zetafunc.

Guest

Law of Restitution

"A small smooth sphere of mass 3 kg moving on a smooth horizontal plane with a speed of 8 m/s collides directly with a sphere of mass 12 kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e."

I am sure I have the method right but I am just getting the wrong sign in my answer.

Textbook answer is e > 0.25. I'm getting e < 0.25...

Diagram:

---> = positive direction

8 m/s            0 m/s
->                 
(3 kg)            (12 kg)
<-                 ->
A m/s           B m/s

By the conservation of momentum:

24 = 12B - 3A
⇒8 = 4B - A

e = (speed of separation)/(speed of approach)

Speed of approach is 8 m/s.
Speed of separation is A + B.

⇒ e = (A + B)/8
⇒ 8e = A + B

So we have:

4B - A = 8
A + B = 8e

Adding both equations gets us:

5B = 8(1 + e)

so e = (5B - 8)/8

B > 0 since moving in positive direction. So e > -1.
A < 0 since moving in negative direction.

Since 8e = A + B and A < 0, we can say:

8e - B < 0

We say that e = (5B - 8)/8, so re-arranging in terms of B, we have B = (8e + 8)/5.

So:

8e - (8e + 8)/5 < 0

⇒ (32e - 8)/5 < 0

⇒ 32e < 8

∴ e < 0.25.

But textbook's answer is e > 0.25. Why?

Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Law of Restitution

hi zetafunc,

Sorry I cannot spend a load of time on this as I've got to go out very shortly.

I think the problem may lie in your restitution formula.

Have a look at

http://en.wikipedia.org/wiki/Coefficient_of_restitution

There's a minus sign there.  Will that swap things round for you?

I'll try and check back this evening (8.25am now).

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

zetafunc

Guest

Re: Law of Restitution

Hi, thanks for the reply. That link was really helpful! My stupid textbook didn't even give this formula.  It showed two objects coming towards each other as having speed of separation v1 + v2, when I thought it would be something like v1 - v2. But the formula in that wiki link makes complete sense. I haven't checked through my work but that's probably where I went wrong! Thanks.

I've been looking through this exercise and I doubt I can even do the last three questions, though. Question 6 says:

"Two identical smooth spheres each of mass m are projected directly towards each other on a smooth horizontal surface. Each sphere has a speed u and the coefficient of restitution between the spheres is e. Show that the collision between the spheres causes a loss of kinetic energy mu²(1 - e²). Describe the 'realistic' situation this could be used to model. State clearly the assumptions you have made."

Given this situation, doesn't this mean the total momentum before and after is 0? I think the speeds of both particles should be the same, too, since they both have the same speed and mass. If this is true, then e = 1, and the collision is perfectly elastic and surely there should be no loss of mechanical energy? I really don't understand this question...

zetafunc

Guest

Re: Law of Restitution

Never mind, solved it

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Law of Restitution

hi zetafunc

I've been travelling to my daughter's during the day.  My wife was a bit disconcerted when I was trying to work out the equations in my head along the M11.

If you imagine that B is a solid brick wall and A hits it with velocity u before and v after the impact, both measured in the same sense, then the formula for an elastic collision is

The minus sign is because A will bounce back away from the wall.

When B is free to move you can modify the formula by making u and v into the relative velocity of A, relative to B.

I would not start with A's velocity towards the left.  If you keep everything as if it were to the right you can consider A < 0 later.

So I had

and

If you eliminate B and make A the subject then you can set up A < 0 and you should get the right inequality.

If e = 0.25 the A = 0

If e < 0.25 then 'bounce' isn't enough to reverse A's direction.

You have to have e > 0.25 for A to bounce back.

Bob

Last edited by Bob (2011-10-28 05:16:57)

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob