# Find inequality for coefficient of restituition

## Homework Statement

A small smooth sphere of mass 3 kg moving on a smooth horizontal plane with speed 8 ms-1 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.

## Homework Equations

v2 - v1/u1 - u2 =e

## The Attempt at a Solution

To be honest, I'm totally lost. I know that v2-v1= 8e, and 0≤e≤1. Also, using principle of conservation of momentum, I know that v1 + 4v2=8. I don't know how to solve this because I only have two equations for 3 unknowns. Any hints?

## Answers and Replies

haruspex
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You have not used the constraint that after collision they move in opposite directions. What inequality does that give you?

Is it v2 - v1 ≥ 8?

haruspex
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Is it v2 - v1 ≥ 8?
No. Which way will the heavier mass move?

No. Which way will the heavier mass move?
The heavier one will move to the right and take a positive velocity.

haruspex
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The heavier one will move to the right and take a positive velocity.
So what does that tell you about how the light mass moves off?

So what does that tell you about how the light mass moves off?
The light mass moves in the opposite direction with a negative velocity

haruspex
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The light mass moves in the opposite direction with a negative velocity
So write that as an inequality.

So write that as an inequality.
I don't know....is the velocity of the lighter ball greater than velocity of heavier ball?
So v1 - v2≥ 0?

I don't know....is the velocity of the lighter ball greater than velocity of heavier ball?
So v1 - v2≥ 0?
And less than 8?

haruspex
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And less than 8?
Just write this statement
The light mass moves ... with a negative velocity
as an inequality.

Just write this statement

as an inequality.
Haha I'm feeling hopeless here. Please forgive my slowness. Is it -v1-v2≤8? I've tried this question for a long time...and I just can't get it.

haruspex
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Haha I'm feeling hopeless here. Please forgive my slowness. Is it -v1-v2≤8? I've tried this question for a long time...and I just can't get it.
The statement I've asked you to write as an inequality does not mention v2, so it will not feature in the inequality.

The statement I've asked you to write as an inequality does not mention v2, so it will not feature in the inequality.
Ok, I have made some headway, but in the right direction or not I'm not sure. So v1≤8. And if so, using the equation from conservation of momentum, I'll get v2≥4. Is it right?

The statement I've asked you to write as an inequality does not mention v2, so it will not feature in the inequality.
Ok, I have made some headway, but in the right direction or not. I'm not sure. So v1≤8. And if so, using the equation from conservation of momentum, I'll get v2≥4. Is it right?

haruspex
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Ok, I have made some headway, but in the right direction or not. I'm not sure. So v1≤8. And if so, using the equation from conservation of momentum, I'll get v2≥4. Is it right?
If you are told x is negative, how would you write that as an inequality?

If you are told x is negative, how would you write that as an inequality?
So it's -v1 ≥ -8 which gives v1 ≥ 8?

haruspex
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So it's -v1 ≥ -8 which gives v1 ≥ 8?
Forget 8, forget about this whole problem for the moment. If you are told that some number x is negative, how do you express that statement as an inequality?

Forget 8, forget about this whole problem for the moment. If you are told that some number x is negative, how do you express that statement as an inequality?
n< 0

haruspex
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n< 0
Right. So apply that to v1 and see where it leads.

Right. So apply that to v1 and see where it leads.
By substituting v1>0, into the equation from conservation of momentum, I get v2>2. Therefore, v2 -v1>2. Using this in coefficient of restitution equation yields e> 1/4

haruspex