Find inequality for coefficient of restituition

In summary, the inequality satisfied by e in this problem is e > 1/4. This is found by taking into account the constraint that the lighter mass moves in the opposite direction with a negative velocity, and using the conservation of momentum equation to find that v2 - v1 > 2. This is then substituted into the coefficient of restitution equation, resulting in e > 1/4.
  • #1
toforfiltum
341
4

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?
 
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  • #2
You have not used the constraint that after collision they move in opposite directions. What inequality does that give you?
 
  • #3
Is it v2 - v1 ≥ 8?
 
  • #4
toforfiltum said:
Is it v2 - v1 ≥ 8?
No. Which way will the heavier mass move?
 
  • #5
haruspex said:
No. Which way will the heavier mass move?
The heavier one will move to the right and take a positive velocity.
 
  • #6
toforfiltum said:
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?
 
  • #7
haruspex said:
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
 
  • #8
toforfiltum said:
The light mass moves in the opposite direction with a negative velocity
So write that as an inequality.
 
  • #9
haruspex said:
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?
 
  • #10
toforfiltum said:
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?
 
  • #11
toforfiltum said:
And less than 8?
Just write this statement
toforfiltum said:
The light mass moves ... with a negative velocity
as an inequality.
 
  • #12
haruspex said:
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.
 
  • #13
toforfiltum said:
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.
 
  • #14
haruspex said:
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?
 
  • #15
haruspex said:
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?
 
  • #16
toforfiltum said:
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?
 
  • #17
haruspex said:
If you are told x is negative, how would you write that as an inequality?
So it's -v1 ≥ -8 which gives v1 ≥ 8?
 
  • #18
toforfiltum said:
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?
 
  • #19
haruspex said:
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
 
  • #20
toforfiltum said:
n< 0
Right. So apply that to v1 and see where it leads.
 
  • #21
haruspex said:
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
 
  • #22
toforfiltum said:
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.
 

FAQ: Find inequality for coefficient of restituition

1. What is the coefficient of restitution?

The coefficient of restitution is a measure of the elasticity or bounciness of a collision between two objects. It is represented by the symbol "e" and is calculated by dividing the velocity of separation by the velocity of approach.

2. How is the coefficient of restitution used in physics?

The coefficient of restitution is used to predict the outcome of collisions between objects. It can help determine how much energy is lost or conserved during a collision and can be used to analyze the motion of objects in various scenarios.

3. How do you find the coefficient of restitution experimentally?

The coefficient of restitution can be found experimentally by measuring the velocities of two objects before and after a collision and plugging those values into the equation e = vsep / vapp, where vsep is the velocity of separation and vapp is the velocity of approach.

4. What does a coefficient of restitution of 1 mean?

A coefficient of restitution of 1 means that the collision between two objects is perfectly elastic, with no energy lost. This means that the objects will bounce off each other with the same velocity and momentum as before the collision.

5. Can the coefficient of restitution be greater than 1?

Yes, the coefficient of restitution can be greater than 1. This would indicate a collision where the objects not only bounce off each other, but also gain additional energy. This is known as a super-elastic collision and is rare in real-world scenarios.

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