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Two spherical, equal density stars, one of mass {m}, radius {R

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Solving time: 5 mins

Two spherical, equal density stars, one of mass $m$ , radius $R$ and another mass $8 m$ are approaching each other for a head-on collision. Initially, when their centres are at distance $d$ , their speeds are negligible. When they collide, determine
(i) the velocity of the smaller star
(ii) distance travelled by the smaller star from initial position.

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IIT JEE Super Course in Physics Mechanics II (Pearson)

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Two spherical, equal density stars, one of mass

m

, radius

R

and another mass

8 m

are approaching each other for a head-on collision. Initially, when their centres are at distance

d

, their speeds are negligible. When they collide, determine
(i) the velocity of the smaller star
(ii) distance travelled by the smaller star from initial position.

12 mins ago

Discuss this question LIVE

12 mins ago

Text solutionVerified

(i) Initial energy of system

= \frac{- Gm 8 m}{d}

Let final velocity of smaller star be v.
Then final velocity of bigger star

= \frac{v}{8}

(

∵

momentum conservation)

∴

Final KE of system

= \frac{1}{2} mv^{2} + \frac{1}{2} 8 m \frac{v ^{2}}{64} = \frac{9}{16} mv^{2}

∵

Radius of bigger star is

2 R

, density same, mass 8 times Final distance between centres

= 3 R

∴

Final

PE = \frac{- G \cdot m \cdot 8 m}{3 R}

Equating energies, \Rightarrow \frac{9}{16} mv^{2} - \frac{8}{3} \frac{Gm ^{2}}{R} = \frac{- 8 Gm ^{2}}{d} v = \frac{8 2}{3 3} \frac{Gm ( d - 3 R )}{Rd}

(ii) Initial position of

CM : mx = 8 m (d - x)

x = \frac{8}{9} d

No displacement for centre of mass.
Final CM:
Final

d

when they touch each other

= 3 R

Position from centre of small star

= \frac{8}{9} (3 R)

No displacement for centre of mass.
Final CM:
Final

d

when they touch each other

= 3 R

Position from centre of small star

= \frac{8}{9} (3 R)

∴

Distance travelled by smaller star

= \frac{8}{9} (d - 3 R) .

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Question Text	Two spherical, equal density stars, one of mass $m$ , radius $R$ and another mass $8 m$ are approaching each other for a head-on collision. Initially, when their centres are at distance $d$ , their speeds are negligible. When they collide, determine (i) the velocity of the smaller star (ii) distance travelled by the smaller star from initial position.
Topic	Gravitation
Subject	Physics
Class	Class 11
Answer Type	Text solution:1
Upvotes	51

Text solutionVerified

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