Collisions

The figure is a two-panel sketch on an $x$–$y$ coordinate grid. The origin marks the point of collision. Before the collision, a blue sphere of mass $m_1$ moves horizontally from the left along the $x$-axis with velocity $\vec{v}_{1i}$. A grey sphere of mass $m_2$ sits stationary at the origin. After the collision, the blue sphere moves up and to the right at an angle $\theta_1$ above the $x$-axis, with final velocity $\vec{v}_{1f}$. The grey sphere moves down and to the right at an angle $\theta_2$ below the $x$-axis, with final velocity $\vec{v}_{2f}$. The arrows representing $\vec{v}_{1f}$ and $\vec{v}_{2f}$ are drawn from the origin, showing the two outgoing paths.

The physical idea is a two-dimensional elastic collision between a moving projectile and a stationary target. The figure makes clear that the motion is confined to a plane — the incoming momentum is entirely along $x$, but after the collision the momentum is shared between both bodies in both $x$ and $y$ directions. The angles $\theta_1$ and $\theta_2$ are measured from the original line of motion (the $x$-axis), and they are not independent: conservation of momentum in the $y$-direction forces a relation between them.

The textbook develops the two fundamental conservation laws from this figure. For an elastic collision, kinetic energy is also conserved. The key equations are:

The first equation is conservation of momentum along the $x$-axis: the initial momentum $m_1 v_{1i}$ equals the sum of the $x$-components of the final momenta. The second equation is conservation of momentum along the $y$-axis: the initial $y$-momentum is zero, so the upward $y$-component of $m_1$'s final momentum must equal the downward $y$-component of $m_2$'s final momentum (the minus sign accounts for opposite directions). The third equation is conservation of kinetic energy, which holds only for an elastic collision.