What Determines the Coefficient of Restitution in a Two-Object Collision?

[Yoruichi]

Homework Statement

As shown in the figure below, small object A (mass: M) collides with small object B (mass: m), which is initially at rest, on top of a horizontal stand fixed to a horizontal floor. Both A and B proceed to shoot horizontally off the stand and fall to the floor. Horizontal distance D from the edge of the stand to the point where A lands is 1/2 of the horizontal distance d to the point where B lands. Friction between the two objects and the stand is negligible.

https://scontent-kul1-1.xx.fbcdn.net/hphotos-xfp1/v/t34.0-12/12767193_1257518357595899_654751098_n.jpg?oh=5ee0294835905cc57ce2d8b8b2eed855&oe=56CC2F3E

What is the coefficient of restitution between A and B?

Homework Equations

The Attempt at a Solution

If I simply substitute M1U1 + M2U2 = M1V1 + M2V2 and the equation above, I will come out with (M - m)/ 2M but that's wrong since I didn't put in the distance factor..

I'm not sure how to bring in the distance factor into the restitution formula. Should I use the v^2 = u^2 + 2as formula?

Also, is it that both A and B are in rest when they land on the floor? If that's the case, won't the final velocity of the object be 0?

 

[ehild]

What do you think which object falls further?
You have a collision and a horizontal projection following it.

Also, is it that both A and B are in rest when they land on the floor? If that's the case, won't the final velocity of the object be 0?

No, you do not know what happens after the objects reach the floor. They can rebound, slide further, but you are interested in the velocity just before impact.
Push a ball from a table, and see what happens when it lands on the floor.

 

[drvrm]

If I simply substitute M1U1 + M2U2 = M1V1 + M2V2 and the equation above, I will come out with (M - m)/ 2M but that's wrong since I didn't put in the distance factor..

well i failed to understand - where you are wrong and what is correct according to you ?
try to explain !

Also, is it that both A and B are in rest when they land on the floor? If that's the case, won't the final velocity of the object be 0?

do you think you are correctly describing the physical situation! check by doing a simple exercise of throwing a ball horizontally from a height h?

 

[Yoruichi]

Thanks for the replies!

How did you get that result?

well i failed to understand - where you are wrong and what is correct according to you ?
try to explain !

Below is my attempt according to the two formulas.

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂
Since U₂ = 0, M₁U₁ = M₁V₁ + M₂V₂

U₁ = (M₁V₁ + M₂V₂) /M₁
V₁ = (M₁U₁ - M₂V₂) / M₁
V₂ = (M₁U₁ - M₁V₁) / M₂

Using the formula

Cr = [ (M₁U₁ - M₁V₁) / M₂) - (M₁U₁ - M₂V₂) / M₁] / [(M₁V₁ + M₂V₂) / M₁]
= {[(M₁M₁U₁ - M₁M₁V₁) / M₁M₂ ] - [M₁M₂U₁ - M₂M₂V₂ / M₁M₂]} x M₁ / (M₁V₁ + M₂V₂)
= { [M₁M₁U₁ - M₁M₁V₁ - M₁M₂U₁+ M₂M₂V₂] / M₁M₂ } x (M₁V₁ + M₂V₂)
= [ M₁M₁U₁ - M₁M₁V₁ - M₁M₂U₁ + M₂M₂V₂ ] / (M₁M₂V₁ + M₂M₂V₂)

*edited
sorry I'm kinda messed up with this..

My bad, after this I realize that this should be the right answer for my attempt.
But still, I doubt that this is the wrong answer. As this formula doesn't require the horizontal distance to come out with the answer... (and I guess this is a bit over complicated?)

No, you do not know what happens after the objects reach the floor. They can rebound, slide further, but you are interested in the velocity just before impact.
Push a ball from a table, and see what happens when it lands on the floor.

do you think you are correctly describing the physical situation! check by doing a simple exercise of throwing a ball horizontally from a height h?

I see, this is my misunderstanding..

What do you think which object falls further?

Object B falls further?

 

[ehild]

Thanks for the replies!

Below is my attempt according to the two formulas.

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂
Since U₂ = 0, M₁U₁ = M₁V₁ + M₂V₂

U₁ = (M₁V₁ + M₂V₂) /M₁
V₁ = (M₁U₁ - M₂V₂) / M₁
V₂ = (M₁U₁ - M₁V₁) / M₂

Using the formula

= [ M₁M₁U₁ - M₁M₁V₁ - M₁M₂U₁ + M₂M₂V₂ ] / (M₁M₂V₁ + M₂M₂V₂)
My bad, after this I realize that this should be the right answer for my attempt.
But still, I doubt that this is the wrong answer. As this formula doesn't require the horizontal distance to come out with the answer... (and I guess this is a bit over complicated?)

It is not an answer. You need to give CR with known quantities, the masses of the objects M and m and the ratio of the horizontal distances they reach. You do not know V1 and V2. It has no sense to make the original formula more complicated by still having the same unknowns.

Object B falls further?

Yes, what does it mean for its velocity after the collision?

 

[Yoruichi]

It is not an answer. You need to give CR with known quantities, the masses of the objects M and m and the ratio of the horizontal distances they reach. You do not know V1 and V2. It has no sense to make the original formula more complicated by still having the same unknowns.
Yes, what does it mean for its velocity after the collision?

I think its velocity after the collision will be faster.

 

[ehild]

I think its velocity after the collision will be faster.

How much faster if it reaches twice the distance then A?

 

[Yoruichi]

How much faster if it reaches twice the distance then A?

The velocity will also be 2 times faster than A, so the ratio of both distance and velocity will be 1:2. Am I right?
So the next step would be substitute the ratio into the formula?

 

[ehild]

The velocity will also be 2 times faster than A, so the ratio of both distance and velocity will be 1:2. Am I right?
So the next step would be substitute the ratio into the formula?

yes, into both formula. First find V_A in terms of the masses and U_A, then substitute both V_A and VB=2V_A into the formula for CR.

 

[Yoruichi]

yes, into both formula. First find V_A in terms of the masses and U_A, then substitute both V_A and VB=2V_A into the formula for CR.

Let V₂ = 2V₁

M₁U₁ = M₁V₁ + M₂V₂
M₁U₁ = M₁V₁ + 2M₂V₁
V₁(M₁+2M₂) = M₁U₁
V₁ = M₁U₁ / (M₁ + 2M₂)

CR = (2V₁ - V₁) / U₁
CR = V₁ / U₁

Sub V₁ = M₁U₁ / (M₁ + 2M₂)
CR = V₁ = M₁U₁ / (M₁ + 2M₂) / U₁
CR = V₁ = M₁U₁ / (M₁ + 2M₂) x (1 / U₁)
CR = M₁ / (M₁+2M₂)

This is the correct answer, thanks a lot! :D

 

[ehild]

You are welcome