Two-Dimensional Collisions, Explosions and Force–Time Graphs

Momentum is conserved one axis at a time

Because momentum is a vector, conservation in two dimensions is two separate equations:

Set up axes, resolve every velocity into components with $v\cos\theta$ and $v\sin\theta$, and never add magnitudes of momenta that point in different directions. A puck moving east that strikes a puck at rest can send both off at angles; the $y$-momenta after the collision must then be equal and opposite because there was no $y$-momentum before. Kinetic energy is a scalar and, if the collision is elastic, gives one more equation with no components at all.

Explosions and recoil

An explosion is a collision run backwards: internal forces push the pieces apart, so total momentum is unchanged. If the system starts at rest, the pieces leave with equal and opposite momenta:

A $4$ kg rifle firing a $10$ g bullet at $400$ m/s recoils at $\frac{0{,}01\times400}{4}=1$ m/s. The kinetic energies are not equal: since $K=\frac{p^2}{2m}$ and both pieces have the same $p$, the lighter piece carries most of the energy — the bullet gets $800$ J, the rifle only $2$ J. Mechanical energy increases in an explosion; it comes from chemical energy, a compressed spring, or a person's muscles.

Force–time graphs

When the force during a collision is not constant, impulse is the area under the $F$–$t$ curve. A triangular force peaking at $600$ N over $0{,}02$ s delivers $\tfrac12(600)(0{,}02)=6$ N·s. The average force is that area divided by the duration, $F_{\text{avg}}=\frac{\Delta p}{\Delta t}=300$ N here — half the peak, as it must be for a triangle. Area below the axis is impulse in the negative direction.