Find inequality for coefficient of restituition

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Homework Help Overview

The problem involves a collision between two spheres of different masses, where one is initially at rest. The objective is to find an inequality for the coefficient of restitution, e, based on the conditions of the collision and the conservation of momentum.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the implications of the spheres moving in opposite directions after the collision and how this affects the inequalities related to their velocities.
  • There are attempts to express the relationship between the velocities of the two spheres as inequalities, with some questioning the direction of motion and the signs of the velocities.
  • Several participants express uncertainty about how to formulate the inequalities correctly based on the given conditions.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem and attempting to derive inequalities related to the velocities of the spheres. Some guidance has been provided regarding the direction of motion and the formulation of inequalities, but there is still a lack of consensus on the correct expressions.

Contextual Notes

Participants note constraints such as the requirement that the coefficient of restitution must satisfy certain conditions (0 ≤ e ≤ 1) and the implications of the conservation of momentum in the context of the collision.

toforfiltum
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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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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?
 
toforfiltum said:
Is it v2 - v1 ≥ 8?
No. Which way will the heavier mass move?
 
haruspex said:
No. Which way will the heavier mass move?
The heavier one will move to the right and take a positive velocity.
 
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?
 
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
 
toforfiltum said:
The light mass moves in the opposite direction with a negative velocity
So write that as an inequality.
 
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.
 

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