# Simultaneous Inelastic Collision

#### Jen

The world is 2D.
I am given three different balls that simultaneously collide.
I know their velocity(both x and y) before impact and two of the balls' initial positions.
The collision is perfectly inelastic and the three balls stopped upon collision.
How do I determine the initial position for the last ball?
I know the time should be the same for all of the balls upon contact, but I don't know what to do.

#### topsquark

Forum Staff
The world is 2D.
Oh that would be nice!
I am given three different balls that simultaneously collide.
I know their velocity(both x and y) before impact and two of the balls' initial positions.
The collision is perfectly inelastic and the three balls stopped upon collision.
How do I determine the initial position for the last ball?
I know the time should be the same for all of the balls upon contact, but I don't know what to do.
Can I presume they all have the same mass so you can calculate the momentum of the two balls before impact?

If so, then you have an equation you can use. Conservation of momentum. The collision is perfectly inelastic and stops all the ball's motion. So you can say there is zero momentum after the collision. Thus you know that the total mometum of the balls is also zero before the collision.

See what you can do with that and let us know if you have any more questions.

-Dan

#### topsquark

Forum Staff
You can use momentum conservation, as I mentioned before. Then you can divide the momentum for the third particle by its mass to get the initial speed. Once you have that then we know there is no acceleration for the mass and we can use
$$\displaystyle x = v_xt$$ and $$\displaystyle y = v_y t$$ to find the initial positions.

-Dan

#### Jen

But I don't know the time, and I don't know the final position so even if I know the displacement how do I find the initial position?

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#### HallsofIvy

Well, what do you know? You say you know the initial positions of two of the balls and their velocity vectors. That is sufficient to determine the point where those two collide and since all three balls collide "simultaneously", that is the point of collision of all three balls. You can use "conservation of momentum" to calculate the momentum (so velocity vector) of the first two balls if they were to collide "inelastically". Since "the three balls stopped upon collision" the momentum vector of the third ball must be the negative of the momentum vector of the first two balls.

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#### topsquark

Forum Staff
Oh, you already posted that one in this thread. I had thought it was going to be a different one.

You know where balls 1 and 2 start. Since you have the velocity of each ball you can say where it is if you know how long it has been in motion. Since the two balls collide (or are at the same coordinates) you can find out where they are when they meet.

For instance, ball 1 has the equation
$$\displaystyle d_{1x} = 1 + (1)t$$

$$\displaystyle d_{1y} = 0 + (0.300)t$$

Find the components for ball 2 and solve for t by equating $$\displaystyle d_{1x} = d_{2x}$$ or you can do the y components.

-Dan