Two neutron stars are separated by a distance of `10^(10) m`. They each have a mass of `10^(30) kg` and a radius of `10^(5) m`. They are initially at rest with respect to each other.
As measured from the rest frame, how fast are they moving when
(a) their separation has decreaed to one - half its initial value,
(b) they are about to collide .

Answer

A, B, D

Explanation

By conservation of momentum, their speeds are same. Using the energy equations we have
`K_(i) + U_(i) = K_(f) + U_(f)`
`0 - (Gmm)/(r_(i)) = 2 ((1)/(2) m nu^(2)) - (Gmm)/(r_(f))`
`:. nu = sqrt(Gm ((1)/(r_(f)) - (1)/(r_(i))))` ..(i)
`= sqrt((Gm)/(r_(i))`
`= sqrt((6.67 xx 10^(-11) xx 10^(30))/(10^(10))`
`= 8.16 xx 10^(4) m//s`
`= 81.6 km//s`

Using Eq, (i), we have
`nu = sqrt(Gm((1)/(r_(f)) - (1)/(r_(i)))`
`= sqrt(6.67 xx 10^(-11) xx 10^(30)((1)/(2 xx 10^(5)) - (1)/(10^(10)))`
`= 1.8 xx 10^(7) m//s = 1.8 xx 10^(4) km//s`