Conservation in two dimensions

Until now collisions have been along a line: a single momentum component to conserve. In two dimensions momentum is a vector with two components, and each is conserved separately. If the collision is also elastic, conservation of kinetic energy is added.

Principle — Equations for an elastic 2D collision

$\begin{aligned} p_{x,A} &= p_{x,B} \quad\text{(conservation of } p_x\text{)} \\ p_{y,A} &= p_{y,B} \quad\text{(conservation of } p_y\text{)} \\ E_\text{cin,A} &= E_\text{cin,B} \quad\text{(elastic)} \end{aligned}$ These are three equations, so at most three unknowns can be determined.

The count of unknowns versus equations is once again the compass. In a 2D elastic collision between two smooth spheres, if one ball is initially at rest and the exit direction of one of the two is known, the three unknowns (for example $v_{1B}$, $v_{2B,x}$, $v_{2B,y}$) are determined from the three conservation equations. The resulting system is not linear — kinetic energy contains the squares of the velocities — but it is solved by substitution.

The physical idea is that resolving along two axes decouples the problem into the two directions: what happens along $x$ is independent of what happens along $y$, provided the axes are chosen carefully. The most convenient choice, as we shall see, is to align one axis with the line joining the centres at the moment of contact.

Connections

Topics: Quantità di moto e urti Concepts: Urto elastico · Conservazione della quantità di moto Methods: Scomposizione in componenti cartesiane

Related exercises: Problema — Urto elastico 2D su un biliardo · Urto elastico tra due biglie · Vero o falso sugli urti