31.6: Collisions in Two Dimensions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Now consider a collision in two dimensions between two masses m1 and m2 (Fig. 31.6.1). Without loss of generality, we can work in a coordinate system that is at rest with respect to mass m2, and in which mass m1 is moving in the +x direction, as shown in the figure. Then before the collision, mass m1 is moving with velocity v1i=v1ii. After the collision, mass m1 moves with velocity v1f=(v1fcosθ1)i−(v1fsinθ1)j; mass m2 moves with velocity v2f=(v2fcosθ2)i+(v2fsinθ2)j.

By conservation of momentum, we know that both the x and y components of the total system momentum are independently conserved. This gives two equations: in the x direction,
pix=pfx
m1v1i=m1v1fcosθ1+m2v2fcosθ2
and in the y direciton,
piy=pfy
0=−m1v1fsinθ1+m2v2fsinθ2.
So Eqs. 31.6.2 and 31.6.4 give us two equations - but in this case there are four unknowns ( v1f,v2f,θ1, and θ2 ). To determine the four unknowns, we need as many equations as we have unknowns, so we're two equations short and we need to provide some more information. For example, if we assume that the collision is perfectly elastic, then we can add another equation, since kinetic energy will be conserved in this case:
Ki=Kf
12m1v21i=12m1v21f+12m2v22f
Now we have three equations (Eqs. 31.6.2, 31.6.4, and 31.6.6), but we still have four unknowns—we still need more information to find the final velocities. To solve the problem, we could be given one of the four unknowns, for example. But the piece of information that's really missing here is the impact parameter of the collision, which is the perpendicular distance between the center of mass m2 and the line along the the initial velocity vector v1i. If the impact parameter is zero, then mass m1 hits mass m2 head-on. If the impact parameter is equal to the sum of the radii of m1 and m2, then the two masses will barely touch in a glancing blow. Knowing the impact parameter is necessary for finding the angles θ1 and θ2.
Collisions in two dimensions are more general that you might think: under a central-force law, motion will be in a plane, so the particles will move in two dimensions. Analyzing two-dimensional collisions of this type is common in particle physics. There the particles typically do not actually touch, but are repelled or attracted by the electrostatic force. The same laws apply in particle physics as what we've described here.