Find inequality for coefficient of restituition

[toforfiltum]

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?

 

[haruspex]

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

 

[toforfiltum]

Is it v_{2 -} v₁ ≥ 8?

 

[haruspex]

Is it v_{2 -} v₁ ≥ 8?

No. Which way will the heavier mass move?

 

[toforfiltum]

No. Which way will the heavier mass move?

The heavier one will move to the right and take a positive velocity.

 

[haruspex]

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?

 

[toforfiltum]

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]

The light mass moves in the opposite direction with a negative velocity

So write that as an inequality.

 

[toforfiltum]

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?

 

[toforfiltum]

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]

And less than 8?

Just write this statement

The light mass moves ... with a negative velocity

as an inequality.

 

[toforfiltum]

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]

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.

 

[toforfiltum]

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?

 

[toforfiltum]

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]

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?

 

[toforfiltum]

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]

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?

 

[toforfiltum]

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]

n< 0

Right. So apply that to v₁ and see where it leads.

 

[toforfiltum]

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]

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

I assume you meant v₁<0, but that answer looks right.