Two stars each of mass M and radius R are approaching each other for a head-on collision. They start approaching each other when their separation is r R. If their speeds at this separation are negligible, the speed v with which they collide would be

Question

Two stars each of mass M and radius R are approaching each other for a head-on collision. They start approaching each other when their separation is r > > R. If their speeds at this separation are negligible, the speed v with which they collide would be (A) v = √GM((1/R)-(1/r)) (B) v = √GM((1/2R)-(1/r)) (C) v = √GM((1/R)+(1/r)) (D) v = √GM((1/2R)+(1/r))

Tardigrade Admin · Accepted Answer

Since the speeds of the stars are negligible when they are at a distance

r

, hence the initial kinetic energy of the system is zero. Therefore, the initial total energy of the system is

E_{i} = K E + PE = 0 + (- \frac{GMM}{r}) = - \frac{G M ^{2}}{r}

where M represents the mass of each star and

r

is initial separation between them.
when two stars collide their centres will be at a distance twice the radius of a star i.e.

2 R

.

Let v be the speed with which two stars collide. Then total energy of the system at the instant of their collision is given by

< b r / > E_{f} = 2 \times (\frac{1}{2} M v^{2}) + (- \frac{GMM}{2 R}) = M v^{2} - \frac{G M ^{2}}{2 R} < b r / >

According to law of conservation of mechanical energy

E_{f} = E_{i}

< b r / > M v^{2} - \frac{G M ^{2}}{2 R} = - \frac{G M ^{2}}{r} or v^{2} = GM (\frac{1}{2 R} - \frac{1}{r}) < b r / >

or

v = GM (\frac{1}{2 R} - \frac{1}{r})

Q. Two stars each of mass $M$ and radius $R$ are approaching each other for a head-on collision. They start approaching each other when their separation is $r >> R$ . If their speeds at this separation are negligible, the speed $v$ with which they collide would be

$v = GM (\frac{1}{R} - \frac{1}{r})$

$v = GM (\frac{1}{2 R} - \frac{1}{r})$

$v = GM (\frac{1}{R} + \frac{1}{r})$

$v = GM (\frac{1}{2 R} + \frac{1}{r})$

Q. Two stars each of mass M and radius R are approaching each other for a head-on collision. They start approaching each other when their separation is r>>R. If their speeds at this separation are negligible, the speed v with which they collide would be

v=GM(R1​−r1​)​

v=GM(2R1​−r1​)​

v=GM(R1​+r1​)​

v=GM(2R1​+r1​)​

Q. Two stars each of mass $M$ and radius $R$ are approaching each other for a head-on collision. They start approaching each other when their separation is $r >> R$ . If their speeds at this separation are negligible, the speed $v$ with which they collide would be

$v = GM (\frac{1}{R} - \frac{1}{r})$

$v = GM (\frac{1}{2 R} - \frac{1}{r})$

$v = GM (\frac{1}{R} + \frac{1}{r})$

$v = GM (\frac{1}{2 R} + \frac{1}{r})$