Two stars each of mass m and radius R approach each other to collide head-on. Initially the stars are at a distance r(R) . Assuming their speeds to be negligible at this distance of separation, the speed with which the stars collide is

Question

Two stars each of mass m and radius R approach each other to collide head-on. Initially the stars are at a distance r(>>R) . Assuming their speeds to be negligible at this distance of separation, the speed with which the stars collide is (A) √G m((1/R)-(1/r)) (B) √G m((1/2 R)-(1/r)) (C) √G m((1/R)+(1/r)) (D) √G m((1/2 R)+(1/r))

Tardigrade Admin · Accepted Answer

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

r

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

E_{i} = K E + PE = 0 + (- \frac{G mm}{r})

...(i)
where

m

represents the mass of each star and

r

the 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

E_{f} = (\frac{1}{2} m v^{2}) \times 2 + (- \frac{G mm}{2 R}) = m v^{2} - \frac{GM m}{2 R}

According to law of conservation of energy,

E_{f} = E_{i}

∴ m v^{2} - \frac{G mm}{2 R} = - \frac{G mm}{r}

or

v^{2} = G m (\frac{1}{2 R} - \frac{1}{r})

or

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

Q. Two stars each of mass $m$ and radius $R$ approach each other to collide head-on. Initially the stars are at a distance $r (>> R) .$ Assuming their speeds to be negligible at this distance of separation, the speed with which the stars collide is

$G m (\frac{1}{R} - \frac{1}{r})$

$G m (\frac{1}{2 R} - \frac{1}{r})$

$G m (\frac{1}{R} + \frac{1}{r})$

$G m (\frac{1}{2 R} + \frac{1}{r})$

Q. Two stars each of mass m and radius R approach each other to collide head-on. Initially the stars are at a distance r(>>R). Assuming their speeds to be negligible at this distance of separation, the speed with which the stars collide is

Gm(R1​−r1​)​

Gm(2R1​−r1​)​

Gm(R1​+r1​)​

Gm(2R1​+r1​)​

Q. Two stars each of mass $m$ and radius $R$ approach each other to collide head-on. Initially the stars are at a distance $r (>> R) .$ Assuming their speeds to be negligible at this distance of separation, the speed with which the stars collide is

$G m (\frac{1}{R} - \frac{1}{r})$

$G m (\frac{1}{2 R} - \frac{1}{r})$

$G m (\frac{1}{R} + \frac{1}{r})$

$G m (\frac{1}{2 R} + \frac{1}{r})$