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An asteroid of mass m is approaching ear...

An asteroid of mass `m` is approaching earth, initially at a distance `10R_(E)` with speed `v_(i)`. It hits earth with a speed `v_(f)` (`R_(E)` and `M_(E)` are radius and mass of earth),. Then

A

`v_(f)^(2)=v_(i)^(2)+(2Gm)/(R_(E))(1+(1)/(10))`

B

`v_(f)^(2)=v_(i)^(2)+(2GM_(E))/(R_(E))(1+(1)/(10))`

C

`v_(f)^(2)=v_(i)^(2)+(2GM_(E))/(R_(E))(1-(1)/(10))`

D

`v_(f)^(2)=v_(i)^(2)+(2Gm)/(R_(E))(1-(1)/(10))`

Text Solution

Verified by Experts

The correct Answer is:
C
(c ) Initial energy of the asteroid is
`E_(i)=K_(i)+U_(i)=(1)/(2)mv_(i)^(2)-(GM_(E)m)/(10R_(E))`
Final energy of the asteroid is
`E_(f)=(1)/(2)mv_(f)^(2)-(GM_(E)m)/(R_(E))`
According to law of conservation of energy, `E_(i)=E_(f)`
`(1)/(2)mv_(i)^(2)-(GM_(E)m)/(10R_(E))=(1)/(2)mv_(f)^(2)-(GM_(E)m)/(R_(E))`
`v_(f)^(2)-(2GM_(E))/(R_(E))=v_(i)^(2)-(2GM_(E))/(10R_(E))`
`v_(f)^(2)=v_(i)^(2)+(2GM_(E))/(R_(E))(1-(1)/(10))`
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