Identical smooth billiard balls of equal mass and combined hard-sphere radius collide…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Momentum, energy and conservation laws · Gravitation and central-force motion
Action and Symmetry, Ch Scattering in a central field
Problem
Identical smooth billiard balls of equal mass m1=m2 and combined hard-sphere radius a=2.0cm collide elastically, with the target ball initially at rest. The centre-of-mass deflection function is χ=π−2arcsin(b/a).
((a)) Find the impact parameter b that gives a centre-of-mass deflection χ=90∘.
((b)) Find χlab corresponding to χ=90∘, and check it against the equal-mass rule χlab=χ/2.
((c)) A stream of 500 balls per second is fired at a sparse array of 50 target balls per square metre. A detector subtending 0.010sr in the laboratory is set at that laboratory angle. Using the equal-mass laboratory cross-section a2cosχlab, find the count rate, and say by what factor it differs from the rate the centre-of-mass value a2/4 would give.