Question

Two neutron stars are separated by a distance of 1.0 x 10^11 m. They each have a mass of 1.0 x 10^28 kg and a radius of 1.0 x 10^3 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

Part A

Now using energy conservation

KEi + PEi = KEf + PEf

Initial Kinetic energy is zero, since initially at rest , So

-G*M*M/d = 0.5*M*V1^2 + 0.5*M*V1^2 - G*M*M/r

r = d/2

-G*M^2/d + 2*G*M^2/d = M*V1^2

G*M^2/d = M*V1^2

V1 = sqrt (G*M/d)

V1 = sqrt (6.67*10^-11*1*10^28/10^11)

V1 = 2582.63 m/sec

Part B

When stars are about to collide

Now using energy conservation

KEi + PEi = KEf + PEf

Initial Kinetic energy is zero, since initially at rest , So

-G*M*M/d = 0.5*M*V2^2 + 0.5*M*V2^2 - G*M*M/r

M*V2^2 = G*M^2/r - G*M^2/d

V2 = sqrt (G*M*(1/r - 1/d))

r = 1*10^3 + 1*10^3 = 2*10^3 m

d = 10^11 m

V2 = sqrt (6.67*10^-11*10^28*(1/(2*10^3) - 1/10^11))

V2 = 18.26*10^6 m/sec

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