How Do Restitution Coefficients Affect Billiard Ball Trajectories?

[aturcotte16]

Homework Statement

A billiard ball B must be shoot into the side hole D by making a rebound on the C-band. Calculate the position x of the impact on the band for the restitution coefficients (e) a) e = 1 and b) e = 0.8.

Relevant Equations

m1v1i + m2v2i = m1v1f + m2v2f
1/2 m1(v1i)2 + 1/2 m2(vi)2 = 1/2 m1(v1f)2 + 1/2 m2 (v2f)2
(v2f)n - (v1f)n = e[ (v1i)n - (v2i)n] (n = normal component)

The problem I have here is that I can't see how the restitution coefficients (e) will change the trajectory of the ball since the speed of the ball B will be on the same direction as the normal component.

And without information one the ball A, I can't see how to resolve that.

I will really apreciate if you can help me on this one. Thanks a lot!

Antony

 

[haruspex]

how the restitution coefficients (e) will change the trajectory of the ball

Ignoring spin, the impulse from the cush is normal to the cush. So what is its affect on the velocity parallel to the cush?

 

[aerobee]

You don't need to know anything about ball A beyond the fact that it was struck by the pool cue exactly as needed to send ball B from ball B's initial position to point C, point C being the exact right spot to have it end up at point D.

The x-location of Point C will vary depending on the restitution coefficient (e), because if (e) is less than 1 the cushion gives the ball a leftward push that affects the ball's vector velocity--changing not just the speed but also the direction ball B travels after the impact.

 

[haruspex]

if (e) is less than 1 the cushion gives the ball a leftward push

That doesn’t sound right. What do you mean? How would it do that?

 

[aerobee]

It’s a kind of friction. The ball is hitting the cushion at an angle. As usual with vectors it’s appropriate to resolve the force of friction into two pieces, one on the y-axis and the other, which is the one we care about, on the x-axis.

 

[haruspex]

It’s a kind of friction. The ball is hitting the cushion at an angle. As usual with vectors it’s appropriate to resolve the force of friction into two pieces, one on the y-axis and the other, which is the one we care about, on the x-axis.

But that has nothing to with coefficient of restitution.
Yes, there is a little friction, and it will cause the ball to acquire some spin, but I believe the question setter intends that to be ignored since no friction coefficient is mentioned.
Also, the friction would be parallel to the cushion, no resolving to be done.

 

[sysprog]

Here's an edited version of your image in which I've tried to show the effect of the cushion having a less-than-perfect elasticity (e<1), so that the rebound point (vertex) is somewhere within the cushion, instead of at the first point of contact, as in the original illustration. Please note that the exit trajectory is consequently, from the shooter's point of view, as @aerobee suggested, left of where it would be if the elasticity were perfect (e=1).

Perhaps the added red angle line will help with your visualization.

 

[aerobee]

I appreciate that (e) reflects physical characteristics like deformation or inelasticity that reduce kinetic energy by reducing velocity. If friction were introduced separately there would be a cumulative effect.

But it’s still fair to talk about (e) being a kind of friction when the impact happens at an angle. (If this were an object hitting the cushion straight on there would be reduced velocity but there would be no x-axis force component.)

 

[aerobee]

It’s Newton’s First Law. If we agree (e) being <1 changes the trajectory then the cushion causes a second force to act on ball B (the first force having been caused by ball A).

 

[haruspex]

the rebound point (vertex) is somewhere within the cushion

Yes, but it is irrelevant in this question. Indeed, it is rather more significant that the lines drawn appear to represent the trajectory of the mass centre, so the rebound point should be shown as one radius from the cush.

the exit trajectory is consequently, from the shooter's point of view, as @aerobee suggested, left of where it would be if the elasticity were perfect

From the shooter's point of view the rebound is to the left. What does it mean to say it is more to the left?
@aerobee seems to be saying the trajectory would be shifted to the left from the cushion's point of view - maybe because s/he's confusing it with friction - and that is certainly not correct.

But it’s still fair to talk about (e) being a kind of friction when the impact happens at an angle.

No it is not, and unless you can turn that into an equation involving e it is not helpful.

I have provided the key fact in post #2.

 

[aerobee]

I’ll take the bait. See Example 1 below.
https://www.brown.edu/Departments/Engineering/Courses/En4/notes_old/oblique/oblique.html

 

[haruspex]

I’ll take the bait. See Example 1 below.
https://www.brown.edu/Departments/Engineering/Courses/En4/notes_old/oblique/oblique.html

where we read that the coefficient of restitution is:
"the ratio of the magnitude of the relative velocity of separation to that of the relative velocity of approach in the normal direction,"
which is the point I was making in post #2.
But I fail to see the connection between the algebra at that link and anything you wrote in posts #3, #5 and #8. Indeed, in post #3 you wrote that the x direction (by which you presumably meant parallel to the cush) is where this "friction like" force acts, whereas your link makes it clear that the imperfect restitution relates to the direction normal to the cush.
Further, this means the post-bounce trajectory will be "flatter", i.e. closer to the cush, whereas you appeared to claim the opposite.

 

[aturcotte16]

You don't need to know anything about ball A beyond the fact that it was struck by the pool cue exactly as needed to send ball B from ball B's initial position to point C, point C being the exact right spot to have it end up at point D.

The x-location of Point C will vary depending on the restitution coefficient (e), because if (e) is less than 1 the cushion gives the ball a leftward push that affects the ball's vector velocity--changing not just the speed but also the direction ball B travels after the impact.

Thanks! Thats exactly where I was wrong. I was looking at this problem for the relation between A and B. But it's more with B and C and where the direction will change with different restitution coefficient.

Thanks for all of you!

..(sorry for my English, I'm french)

 

[haruspex]

Thanks! Thats exactly where I was wrong. I was looking at this problem for the relation between A and B. But it's more with B and C and where the direction will change with different restitution coefficient.

Thanks for all of you!

..(sorry for my English, I'm french)

Given the conflicting advice you have had regarding how e<1 affects the trajectory, please post the answers you obtained.

 

[sysprog]

So, @aturcotte16 , you've indicated (by reacting to it with a 'like') that you've seen the most recent post in this thread by @haruspex -- will you please, if you can, respond with the answer(s) you've arrived at for e<1, or if you can't yet do that, let us know what may be presenting a difficulty for you? Thanks.

 

[aturcotte16]

please post the answers you obtained.

Ok, so at first, I didnt see the problem the right way. It's more a geometric problem. Here is my solution, sorry it's in french..

When e is 1, we can consider the ''similar triangle'' to find that x = d/3
When e is 0.8, we have to find the ''dimension'' of the others triangle, I found that since 0.8 influence only the normal speed, the geometry of the seconds triangles will only change it's normal side by 0.8, so the tangent side is x/0.8

Reall sorry for my English!

Here is the solution I found

 

[sysprog]

Vraiment, votre Anglais n'est pas trop pauvre, et regardement votre probleme: vous avez prouvé qu'il fallait changer les coussins (bandes) de la table.

[Really, your English is not too poor, and regarding your problem, you have proven that it will be necessary to change the cushions on the table.]

Thanks for responding with the answer and explanation you found.

 

[haruspex]

Ok, so at first, I didnt see the problem the right way. It's more a geometric problem. Here is my solution, sorry it's in french..

When e is 1, we can consider the ''similar triangle'' to find that x = d/3
When e is 0.8, we have to find the ''dimension'' of the others triangle, I found that since 0.8 influence only the normal speed, the geometry of the seconds triangles will only change it's normal side by 0.8, so the tangent side is x/0.8

Reall sorry for my English!

Here is the solution I foundView attachment 241535

Looks good. Well done.