# Momentum and Energy problem (ball and incline collision)

1. Dec 18, 2009

### Idoke

1. The problem statement, all variables and given/known data
A ball, with mass $$m_{1}=6 kg$$ and speed $$v_{0}$$ (unknown) hits an inclined plane (as shown) of mass $$m_{2}=42 kg$$ at rest on a frictionless floor.
there is no friction between the ball and the plane.
the angle $$\beta$$ is 10 degrees.
Question: what is the value of $$\alpha$$?

2. Diagram

3. The attempt at a solution
I used three things:
1. The conservation of momentum parallel to the X axis, because there are no forces on the system in that axis:
$$m_{1}v_{0}=m_{1}v_{1}cos\beta+m_{2}v_{2}$$
where $$v_{1}$$ and $$v_{2}$$ are the velocities after the collision of the ball and plane, respectively.
EDITED: the equation was wrong, the inclined plane had no velocity before the collision.

2. The conservation of the ball's momentum on the axis parallel to the inclined plane (because no forces are acting on the ball in that axis) :
$$m_{1}v_{0}sin\alpha=m_{1}v_{1}sin\beta$$

3. Conservation of kinetic energy:
$$1/2m_{1}v_{0}^{2}=1/2m_{1}v_{1}^{2}+1/2m_{2}v_{2}^{2}$$

Now I have a problem, I have 3 equations but 4 unknowns (v1, v2, alpha and v0). could someone shed some light on this?
I thought about doing this in the center of mass reference frame but i always get stuck with the fact that the momentum is not conserved in the Y direction.
help would be appreciated.
Thanks.

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Last edited: Dec 19, 2009
2. Dec 19, 2009

### ehild

You can eliminate v2 from the equations for the energy and horizontal momentum, and than solve for v1/v0.

ehild

3. Dec 19, 2009

### Idoke

But I still need to find alpha!
Do I not have the same predicament?
I feel like I'm missing another equation, or there might be a more elegant solution than this mess.
If I could find how much impulse (or change in momentum) the floor had on the ball and plane I think I could arrive at a solution.
What do you think?

4. Dec 19, 2009

### denverdoc

from your second eqn V1/V0=sin (alpha)/sin (beta); the latter is known.

5. Dec 19, 2009

### Idoke

First of all, thanks for your time, both of you.
Now, I fixed the first equation, the inclined plane had no velocity before the collision.
I might be really dense, but I don't understand how to eliminate v2 from my equation or how that helps me solve the system.
If you could clarify, i'd be very grateful.
Ido.

6. Dec 19, 2009

### denverdoc

I believe what ehild was suggesting is using eqns 1 & 3 to get the ratio v1/vo

in other words: use eqn 1 to express v2 in terms of masses, angles and v1

something like v2=m1/m2 * (1-cos(alpha)) v1. Then use your conservation of energy formula but substitute for v2 the eqn above. You should be able to get a v1/vo ratio from that. Then use eqn 2 as I mentioned.

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