# Ball collision momentum problem

1. Jun 20, 2014

### petoknm

1. The problem statement, all variables and given/known data

Hello. I was solving this problem about two balls in a plane colliding elastically. The first one had mass 100g radius 5cm and started with velocity of 3m/s to the right and hit the second ball with mass 50g radius 3cm(initially stationary). It hit the second ball such that the line directed by the velocity vector of the first ball going through the center of the first ball and the line parallel to it going through the center of the second ball (these two lines) are separated by a distance of 1cm.

2. Relevant equations

Conservation of energy AND conservation of momentum

3. The attempt at a solution

Well I started to identify the angles after the collision and I came up with the these angles:
first ball:
cos(alpha')=1/8; 0<=alpha'<=pi/2
and the second ball:
sin(alpha)=1/8; 0<=alpha<=pi/2
where alpha' is the angle above the x-axis to the velocity vector of the first ball and alpha is the angle below the x-axis to the velocity vector of the second ball.
Now all we need are the magnitudes of the velocity vectors after the collision. Because it is an elastic collision the energy is conserved and the momentum is conserved. So for energy we have
18=2|v1|^2+|v2|^2
And for the momentum we have
2*v0=2*v1+v2
2<3;0>=2a<cos(alpha');sin(alpha')>+b<cos(alpha);-sin(alpha)>
where a,b are the magnitudes of the vectors.

But this system has no solution. Where is the problem? Thank you!

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Last edited: Jun 20, 2014
2. Jun 21, 2014

### Simon Bridge

It is best practice to do the algebra first and put the numbers in as late as you can.
Why is there no solution?

i.e. do you have more unknowns than you have equations?

Aside: if the problem specifies that these are balls - are they rolling without slipping?
Do you need to account for conservation of angular momentum as well?
Or are these really just a couple of circular objects sliding on a table?

3. Jun 21, 2014

### petoknm

They are sliding. And I have one standard equation (energy conservation) and one two dimensional vector equation (momentum conservation) so basically I have three equations and just two unknowns and therefore I'm not able to find a solution.

4. Jun 21, 2014

### Simon Bridge

Three questions and two unknowns means that one of the equations is surplus to requirements.

5. Jun 21, 2014

### petoknm

But I think that both energy and momentum are conserved in this situation... Or am I wrong?...

6. Jun 21, 2014

### bobie

You should start finding the angle of impact, which is the angle between the direction of the vector (the blue line in your picture) and the line joining the centres of the sphere.
After the collision ball B will move in that direction

7. Jun 21, 2014

### petoknm

That's exactly what I did...I imagined the situation at the moment of impact... There is a right triangle with hypotenuse 8 and opposite side 1...

8. Jun 21, 2014

### bobie

and what is the angle of impact?

9. Jun 21, 2014

### petoknm

Sin(alpha)=1/8...alpha~7deg

10. Jun 21, 2014

### bobie

You need the cosine .992156 to find the x component of velocity.
then you know P = .1*3 and Ke =.45 and M/m =2, that's all you need.

P =.3 = M*vM*cos α + mvm*cos 7.18
KE = .45 = M*vM2/2 + m*vm2/2

hint: m will go at 7.18°, v= 1.9843. (since v0=M+m, v=2cos7°)

Last edited: Jun 21, 2014
11. Jun 21, 2014

### Simon Bridge

@bobie: this is what was done in post #1 - last equation.
@petoknm: the original problem statement does not claim that energy and momentum are conserved, no.
However, I have been advising you on the assumption that the collision is elastic.
Have you tried to use two of the three equations to get a solution, or did you just stop when your realized you had three equations and only two unknowns?

12. Jun 21, 2014

### bobie

Hi Simon, is that the equation you are referring to? I couldn't and cannot read it , and other lines.
I couldn't get what is his problem.

It seems he is assuming the collision is elastic.

Last edited: Jun 21, 2014
13. Jun 21, 2014

### Simon Bridge

That's right - the equation is hard to read - <a;b> is a vector (a,b)t
Yes - OP is assuming the collision is elastic. Also assuming that the objects move on perpendicular trajectories after the collision.

I agree that it is very straight forward, I am trying to get OP to do the next step.
There are more equations than unknowns - so the next step is to solve the simultaneous equations.
The results will say more and allow OP to troubleshoot the answer further.

14. Jun 21, 2014

### bobie

Didn't get that, if that's true, we must warn him that it is a gross mistake, probably that is the cause.

15. Jun 21, 2014

### Simon Bridge

It'll come out in the wash.