# 2D Collision Problem: Given 2 masses, a velocity, and an angle, find another angle and a velocity

I was given the mass and velocity of an object, and I know that it is traveling in a straight line along the x-axis (so I suppose I've been given a second angle). It collides with an object with known mass at rest. The first object rebounds at a given angle. What is the velocity (magnitude and direction) of the second object after the collision? You can't assume that it's an elastic collision.

I started with two component forms of the conservation of momentum equation, and reduced to three variables by replacing $Ax'$ with $A'cos(\theta A')$, and so on for each component. But now I have three variables and two equations! I've racked my brain, and I can't think of what to use as a third equation. What should I do?

• Conservation of kinetic energy as the collision is elastic. – Farcher Nov 1 '16 at 23:18
• I said at the end of the first paragraph that you can't assume it's elastic. – RothX Nov 1 '16 at 23:26
• The problem has no unique solution if you don't know that the collision is elastic, you would need not just the angle of the first object, but also its velocity. You can tell because you know the initial and final momentum (same as initial), but by letting the velocity of the first object be a variable, you get a range of momentum and energy for the second object. That's not a unique solution. – Ken G Nov 1 '16 at 23:50
• @KenG Well, I have the initial velocity, but not the final. Is that what you mean? – RothX Nov 1 '16 at 23:53
• @KenG That should probably be posted as an answer – David Z Nov 1 '16 at 23:55

OK, it seems to me the answer is there is something wrong with the question. You either need to know the collision is elastic, so you know the total energy, or you need to know the full momentum of the first particle, so that you can know the momentum of the second particle. Otherwise, there are a range of velocities for the first particle that will yield different solutions for the second particle, where the kinetic energy is the variable.

You have 2 unknowns (magnitude and direction of velocity for 2nd object) but only 1 equation (conservation of linear momentum). The problem cannot be solved. As you are aware, you don't have enough equations or 'conditions'. You can use conservation of energy if you know that KE is conserved, or what % is conserved. But you cannot create another equation without another 'condition'.

So far we are assuming that the objects are point masses. Collisions between point masses are uncertain because we don't know the direction along which contact is made. If the particles are circles or spheres, and if you are given information about their radii and the perpendicular distance of the 2nd centre from the path of the 1st object, then you can work out the direction along which contact is made and momentum is transferred. This provides an extra condition, enabling you to solve the problem.