Momentum in Two Dimensions

A collision can turn a ball around a corner without slowing it down, because speed only cares about magnitude while momentum cares about direction too. Changing direction means changing the velocity vector, even if its length stays fixed, and that requires an impulse pointed somewhere other than along the original path. Force and impulse are vectors, so they can redirect motion without doing anything to its magnitude.

This is why you split momentum into an x column and a y column and treat them as two separate bookkeeping problems that never mix. The total momentum vector is just p = p_x i + p_y j, and Newton's second law in impulse form, J = delta p, applies component by component: J_x = m(vx_f - vx_i) and J_y = m(vy_f - vy_i). You solve each axis as its own one-dimensional problem, then combine the results with the Pythagorean theorem and an angle if you need the full vector answer.

The same logic drives conservation of momentum in two dimensions. A direction's momentum is conserved only if there is no external force acting along that direction; if a force acts along x, the x momentum changes even if y momentum is untouched. This is exactly the projectile motion case: horizontal momentum survives because nothing pushes horizontally, while vertical momentum does not because gravity is a real external force along y. The same splitting applies to a ball bouncing off a wall at an angle, where the wall's normal force flips the perpendicular component of momentum but leaves the parallel component alone.