It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a problem, the momentum vectors can be, and usually will be, expressed in component form. Conservation of momentum is valid in each direction independently.

The method for solving a two-dimensional (or even three-dimensional) conservation of momentum problem is generally the same as the method for solving a one-dimensional problem, except that the momentum is conserved in both (or all three) dimensions simultaneously. The following steps are carried out to solve a momentum conservation problem in multiple dimensions:

1. Identify the closed system.
2. Write down the equation representing the conservation of momentum in the x-direction, and solve it for the desired quantity. When calculating a vector quantity (velocity, usually), this will give the x-component of the vector.
3. Write down the equation representing the conservation of momentum in the y-direction, and solve. This will give the y-component of the vector quantity.
4. Similar to calculating for a vector quantity, apply the Pythagorean theorem to calculate the magnitude, using the results of steps 2 and 3.

Two-dimensional collision experiments have revealed much of what we know about subatomic particles, as seen in medical applications of nuclear physics and particle physics. For instance, Ernest Rutherford discovered the nature of the atomic nucleus from such experiments.

This text is adapted from Openstax, University Physics Volume 1, Section 9.5: Collisions in Multiple Dimensions.