Variation on 3-ball elastic collision

[epilepticbunny]

Homework Statement

hello! so i am trying to figure out how to calculate the resultant velocities and directions(angles/vectors) that two perfectly elastic spheres might travel in if they were to be hit simultaneously by a third sphere at an angle. all the spheres are of equal mass, initial velocity isn't important as long as all balls are moving after the collision.
ah and assuming that the two stationary balls are parallel to the x-axis, with the y-axis between the 2 balls, the third ball could come at any angle between 30 to 0 degrees to the vertical.. and contact the 2 balls simultaneously..

Homework Equations

V1f^2 + V2f^2 + V3f^2 = V1i^2
(since all the spheres are of equal mass)

and conservation of momentum should be

V1i = V1f + V2f + V3f i think...

The Attempt at a Solution

i have read the other relevant forum posts and i know how to calculate the resultant forces and directions if the collision is head-on, like o-->8 but not if the first ball hits the other two balls at an angle. i would be really grateful if anyone could help me out with this issue.

 

[mfb]

The same rules as for two-body collisions still apply: momentum transfer is orthogonal to the contact. In your specific case, it is easy to get the contact directions, as the three balls form an equilateral triangle (all spheres have the same radius?).
There won't be momentum transfer between the two balls initially at rest. That gives you a total of 5 equations for 6 unknowns, so there is still some freedom left in the process.

 

[epilepticbunny]

The same rules as for two-body collisions still apply: momentum transfer is orthogonal to the contact. In your specific case, it is easy to get the contact directions, as the three balls form an equilateral triangle (all spheres have the same radius?).
There won't be momentum transfer between the two balls initially at rest. That gives you a total of 5 equations for 6 unknowns, so there is still some freedom left in the process.

ah... sorry but i don't really understand which 5 equations I'm supposed to get...
i'm assuming hte 6 unknowns are V1 to V3 (final), x and y for each.
but i only get 4 equations,
v_1x+v_2x+v_3x=v_1i
and the same thing for the y values
then the simplified conservation of K.E. for x and y values.
(yes all spheres have the same radius)
so sorry, can't seem to wrap my head around it...

 

[mfb]

i'm assuming hte 6 unknowns are V1 to V3 (final), x and y for each.

Right.

Kinetic energy is only one equation, there are no "x energy" and "y energy".

The direction of momentum transfer 1->2 and 1->3 give one equation each.

 

[epilepticbunny]

Right.

Kinetic energy is only one equation, there are no "x energy" and "y energy".

The direction of momentum transfer 1->2 and 1->3 give one equation each.

does that mean the two balls would always go in the same direction regardless of the angle the 3rd ball hits them from and only the direction of the 3rd ball would change with the angle?

 

[mfb]

At least if you neglect things like a rotation of balls, deformations, friction and so on: yes.

 

[epilepticbunny]

At least if you neglect things like a rotation of balls, deformations, friction and so on: yes.

sorry, i still can't seem to get the equations out...

 

[mfb]

You know the direction of motion, that is a relation between the x and y components of your velocity.