KEAM 2015 Physics and Chemistry Paper

since, the initial velocity of body is zero, its total energy is
E=‌

−GmM

I

....(i)
where, m is the mass of body, M is the mass of the Earth and I its distance from the centre of the Earth. When the body reaches the Earth, let its, velocity be v and its distance from the centre of the Earth equals the Earths radius R. Therefore, the energy is
E=‌

1

2

mv2−‌

GMm

R

....(ii)
Equating Eqs. (i) and (ii), we get
v2=2GM(‌

1

R

−‌

1

I

)
Also, g=‌

GM

R2

. Therefore, GM=gR2.
Using this in the above equation, we get
v=R[2g(‌

1

R

−‌

1

r

)]1/2
Now, r=2R (given). Therefore,
v=R[2g(‌

1

R

−‌

1

2R

)]1/2
⇒ Velocity of a body which strike the Earth's surface, v=√gR