Two stars each of mass M are approaching each other for a head-on collision. When they are at a distance r, their speeds are negligible. The radius of both stars is R (rR) . What is the speed with which they collide?

Question

Two stars each of mass M are approaching each other for a head-on collision. When they are at a distance r, their speeds are negligible. The radius of both stars is R (r>>R) . What is the speed with which they collide? (A) ((G M/R))(1/2) (B) ((G M/2 R))(1/2) (C) ((G M/4 R))(1/2) (D) ((2 G M/R))(1/2)

Tardigrade Admin · Accepted Answer

Ui+Ki=Uf+Kf (-G M2/r)+0=(-G M2/2 R)+2((1/2) M v2) G M2[(1/2 R)-(1/r)]=M v2 ⇒ v2=(G M(r-2 R)/2 R r) As (r>>R),(r-2 R)=r ⇒ v=((G M/2 R))(1/2)

Q. Two stars each of mass $M$ are approaching each other for a head-on collision. When they are at a distance $r$ , their speeds are negligible. The radius of both stars is $R$ $(r >> R) .$ What is the speed with which they collide?

$(\frac{GM}{R})^{\frac{1}{2}}$

$(\frac{GM}{2 R})^{\frac{1}{2}}$

$(\frac{GM}{4 R})^{\frac{1}{2}}$

$(\frac{2 GM}{R})^{\frac{1}{2}}$

$U_{i} + K_{i} = U_{f} + K_{f}$
$\frac{- G M ^{2}}{r} + 0 = \frac{- G M ^{2}}{2 R} + 2 (\frac{1}{2} M v^{2})$
$G M^{2} [\frac{1}{2 R} - \frac{1}{r}] = M v^{2} \Rightarrow v^{2} = \frac{GM ( r - 2 R )}{2 R r}$
As $(r >> R), (r - 2 R) = r \Rightarrow v = (\frac{GM}{2 R})^{\frac{1}{2}}$

Q. Two stars each of mass M are approaching each other for a head-on collision. When they are at a distance r, their speeds are negligible. The radius of both stars is R (r>>R). What is the speed with which they collide?

(RGM​)21​

(2RGM​)21​

(4RGM​)21​

(R2GM​)21​