Two uniform spherical stars made of same material have radii `R` and `2R`. Mass of the smaller planet is `m`. They start moving from rest towards each from a large distance under mutual force of gravity. The collision between the starts is inelastic with coefficient of restitution `1//2`.
Kinetic energy of the system just after the collision is

Options

A. `(8Gm^(2))/(3R)`
B. `(2Gm^(2))/(3R)`
C. `(4Gm^(2))/(3R)`
D. cannot be determined

Answer

b

Explanation

From conservation of linear momentum velocity before collision `vec(P)_(1)+vec(P)_(2)=0impliesP_(1)=P_(2)`
`m u_(1)+8m u_(2)=0impliesu_(1)=-8u_(2)...(1)`
For kinetic energy
from energy conservation
`U_(i)+K_(i)=U_(f)+K_(f)`
`0+U=-(G(m)(8m))/(3R)+K_(f)impliesK_(f)=8/3(Gm^(2))/R`
This is the kinetic energy before collision.
For kinetic energy after collision from conservation of linear momentum.
`0=[m_(1)v_(1)+m_(2)v_(2)impliesv_(1)+8v_(2)=0....(2)`
`e=("velocity of separation")/(velocity of approach")`
`1/2=(v_(2)-v_(1))....(3)`
From `(2)` and `(3) 9v_(2)=(9u_(1))/16`
`v_(2)=(9u_(1))/16` and `v_(1)=(-u_(1))/2` or
`v_(2)=-(u_(2))/2` and `v_(1)=-(u_(1))/2`
It means total be `KE` after collision `=1/4xx` total `KE` before collision
`=1/4xx8/3(Gm^(2))/R=2/3(Gm^(2))/R`