# Two-body elastic collisions in two dimensions

I have a problem in which two objects collide (elastic) in two dimensions. I am given the vectorial components of the initial velocities of each object as well as their masses, which are different. I also know the final velocity of the smaller object, which is 0 (it comes to rest after the collision). I am to find the vectorial components of the final velocity of the larger object, assuming complete conservation of momentum.

The way I attempted this problem is by separating it into two one-dimensional problems, one for the X-axis and the other for the Y-axis, applying the proper vectorial components of the velocity to each problem, and the solution to them would be their respective vectorial components of the velocity of the larger object after the collision.

Is this the correct way to do this?

Edit: Well, I just calculated the total momentum of my solution, and it's nothing like the total momentum of the two objects prior to the collision. This isn't looking good.

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## Answers and Replies

Is this a real life problem or a textbook problem?
For your momentums to work out, you'll need the velocity of the second object immediately after the collision. It cant have 0 velocity after being hit unless it is acted on by another external force.

whozum said:
Is this a real life problem or a textbook problem?
For your momentums to work out, you'll need the velocity of the second object immediately after the collision. It cant have 0 velocity after being hit unless it is acted on by another external force.

It's a textbook problem. It doesn't make much sense to me how the smaller mass would come to rest either, but I gotta run with it I guess.

Incidentally, it would appear I misread the problem. I thought I had to find the velocity vector of the larger object after the collision, but instead I'm supposed to find its kinetic energy. That makes things much more simple. Still, though, I think the knowledge as to what I would have needed to do to find the velocity vector is to be desired.

The only case I know when one object has no velocity after elastically colliding with another is when both objects have the same mass and they collide head on. In this case, the colliding object stops, the object hit goes off with the speed of the object which just hit it. Obviously momentum and and energy are conserved. That this is not just a textbook answer can be readily ascertained by actually doing it with either pool balls or shuffleboard pucks.

Another way to look at it. A two dim problem like this has 4 unknowns given the pre-collision velocities and masses. Conservation of energy gives you one equation and conservation of momentum gives you two additional equations. That's 4 unknowns and 3 equations which is not solvable. You however are given two more equations and that's more info than needed. Hence you can dispense with conservation of energy and solve the prob strictly from conservation of momentum. After you finish, you might find it interesting to calculate the kinetic energy of the particles before and after the collision.