The impact parameter

For an elastic collision between two smooth spheres there is a powerful geometric simplification. During contact the only force is the normal reaction along the line joining the centres: with no friction, there can be no tangential force. This completely fixes the direction of the momentum exchange.

Principle — Impact parameter

The impact parameter $b$ is the distance between the centre of the struck ball and the line of motion of the striking ball. If the two spheres have radii $r_1$ and $r_2$: $\sin\alpha = \frac{b}{r_1 + r_2}$ where $\alpha$ is the angle between the direction of the striking ball and the line joining the centres at the moment of impact.

Since the force is along the line joining the centres, if the velocities are resolved along the $x$ direction (the joining line) and $y$ (perpendicular), the $y$ component of each ball does not change: $v_{1A,y} = v_{1B,y}$ and $v_{2A,y} = v_{2B,y}$.

The impact parameter $b$ measures by how much the trajectory of the striking ball “misses” the centre of the struck ball. At the moment of contact the line joining the centres forms an angle $\alpha$ such that $\sin\alpha = b/(r_1+r_2)$.

The value of $b$ decides the type of collision: if $b = 0$ the collision is central (head-on, as in 1D); if $b$ is close to $r_1 + r_2$ the collision is nearly glancing and the balls deflect only slightly. Knowing $b$ means knowing the geometry of the impact, and hence the direction along which momentum is transferred — the missing piece of information that closes the system of three conservation equations.

Connections

Topics: Quantità di moto e urti Concepts: Urto elastico Skills: Uso della trigonometria

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