Pool ball hits wall

Anony-mouse

Hello, I have a question.
I have a pool ball mass M and radius R, that hits the pool table wall, at speed V0, angular speed W0 at an angle alfa0, and then goes away with an angle alfa1, speed V1 and angular speed W1.

If the ball wasn't spinning then I think it would be easy to solve with linear algebra, but since it can be spinning in weird ways (W0 is a 3d vector), I think the equations involve the friction with the wall, and that is beyond my expertise. Can anyone help me?

Lojzek

What is known and what must be calculated? I would suppose that the parameters after the
recoil are unknown, but you listed them among the known data?

Anony-mouse

Oh, I'm sorry. Yes, I know M, R, V0 and W0, and I need to know V1 and W1. (I don´t know alfa0, but it should be easily calculated using V0 and the normal to the wall right?)

edit: Oh, I would show what I've done so far, but I have no idea where to start. Any pointers would be appreciated.

DaveC426913

This is why there are guidelines about the format of homework questions. You should read em and follow em.

Anony-mouse

I'm sorry. Here is the reformulated question:

I have a pool ball mass M and radius R, that hits the pool table wall, at speed V0, angular speed W0 at an angle alfa0, and then goes away with an angle alfa1, speed V1 and angular speed W1.

Things I know: M, R, V0 and W0. I also know the normal to the wall (N). I'm guessing since it's a pool ball, that the collision is elastic.
Things I want to know: V1 and W1.

2. Relevant equations
If the ball wasn't spinning then I think it would be easy to solve with linear algebra, but since it can be spinning in weird ways (W0 is a 3d vector), I think the equations involve the Friction with the wall, and that is beyond my expertise. Can anyone help me with the equations I need to use?

3. The attempt at a solution
I can already calculate alfa0 using V1 and N, but that's it.

Hootenanny

If the collision is elastic (i.e. the wall does not deform and the ball does not slip) then what do you know about the total energy of the collision?

Anony-mouse

The total energy should stay the same. Some energy of the rotation should be lost because of friction, but I don't know where it would go (to the velocity of the ball?).

Anony-mouse

Oh, I just found out that I didn't see something. The wall is axis aligned, so it should really simplify the equations.

Anony-mouse

Ok, so this is what I got. I need to have the same energy before and after the collision. But some of the angular velocity will be lost because of friction with the wall. So first I need to see what the angular velocity is after the collision, and then I can use something like this right?
[tex]
\ m v_0^2 + I {\omega_0}^2 = m \underline{v_1}^2 + I {\omega_1}^2
[/tex]
Where I is the moment of inertia of a sphere. From this I can calculate V1. Is this right?
What I still need to figure out is how to calculate the angular velocity lost because of friction. Any pointers?

Hootenanny

You're looking good so far. What can you say about the [perpendicular/parallel] components of the linear velocity before and after the collision? Do they both change?

Anony-mouse

The perpendicular component would have to change, because otherwise the ball would go right through the wall, so I'm guessing the parallel component stays the same?
So, let's say the wall is horizontal. The y component would change, while the x component would stay the same.
Then, I can take the equation from my last post and split [tex]v_1[/tex] in two. So then I have one equation with 2 (or 3) variables to solve. [tex]v_1_y[/tex] and [tex]{\omega_1}^2[/tex]
So I only need to know what [tex]{\omega_1}[/tex] is and I can calculate [tex]v_1_y[/tex] from that.

Hootenanny

Perhaps my previous hint was a little misleading. Both components of the velocity do change, however, since the only force acting in the perpendicular direction is the normal reaction force, the magnitude of the perpendicular component of the velocity doesn't change, but obviously it's direction is reversed. The magnitude of the parallel component, however, does change.

Anony-mouse

Ok, so since [tex] v_1_y = -v_0_y [/tex], now I have
[tex]
\ m v_0^2 + I {\omega_0}^2 = m(\underline{v_1_x^2} - v_0_y^2) + I \underline{{\omega_1}^2}
[/tex]
I have underlined the ones I don't know.

Hootenanny

Good. So when the ball loses some angular velocity, something else must increase to conserve energy.

Anony-mouse

Yes, [tex]v_1_x[/tex] should increase.
So to calculate it, I can use this equation:
[tex]
\sqrt{(m v_0^2 + I {\omega_0}^2 - I \underline{{\omega_1}^2})/m + v_0_y^2} = \underline{|v_1_x|}
[/tex]

Now, to calculate [tex]\omega_1[/tex], I should take the perpendicular velocity and multiply it by the ball mass and some friction coefficient to get how much is [tex]\Delta\omega[/tex] right?

Hootenanny

Looking good

The equation would be much easier to solve if you could relate the final angular velocity to the final linear velocity ...

Lojzek

It is possible to obtain another simple equation connecting the change of parallel velocity and the change of angular velocity during the collision.
Hint: although you don't know the friction force and how long it acts, you do know that it
causes both acceleration and angular acceleration, because it does not act on the center of mass, but a distance R away (so it creates a torque).