Question

Two neutron stars are separated by a distance of 1.0 x 10¹⁰ m. They each have a mass of 1.0 x 10³¹ kg and a radius of 1.0 x 10⁵ m. They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving when (a) their separation has decreased to one-half its initial value and (b) they are about to collide?

Answer 1

initial GPE = -GmM / r = -6.674e?11N·m²/kg² * (1e31kg)² / 1e10m

initial GPE = -6.674e41 J

(a) When their separation is halved, their GPE is doubled; the difference must be the KE (which is shared by the two stars; so

KE = 6.674e41 J = 2*½mv² = 1e31kg * v²

v = 2.58 x 10⁵ m/s

(b) When they are about to collide they are separated by 2 radii, and

GPE = -6.674e?11N·m²/kg² * (1e31kg)² / 1e5m = -6.674e46 J

Since the initial GPE is 6 orders of magnitude smaller, the shared KE is now

KE ? 6.674e46 J = 2 * ½ * 1e31kg * v²

v = 1.16 x 10⁸ m/s