Question

the height from the surface of earth at which the total energy of the satellite is equal to it's potential energy at a height 2R from the surface of earth is (R=radius of earth)

a)R/2 b)R c)R/4 d)3R/4

Answer 1

Potential at height h from the Earth's surface is given by $\mathbf{U=-\frac{GM}{(R+h)}}$
Where M is the mass of satellite and R is the radius of earth.
Now, h = 2R [ according to question ]
Potential energy , U = - GM/(R + 2R) = -GM/3R

We also know, total energy = 1/2 potential energy
e.g., Total energy = $\mathbf{U=-\frac{GM}{2(R+h')}}$

A/C to question,
Total energy at h' height = potential energy at h height
$\mathbf{U=-\frac{GM}{2(R+h)}}$ = -GM/3R
⇒-GM/2(R + h) = -GM/3R
⇒2(R + h) = 3R
⇒2R + 2h = 3R
⇒h = R/2

Hence, at h = R/2 height total energy of satellite is equal to potential energy of satellite at 2R .
∴ option (a) is correct

Answer 2

= -GM/3R

⇒-GM/2(R + h) = -GM/3R

⇒2(R + h) = 3R

⇒2R + 2h = 3R

⇒h = R/2

Hence, at h = R/2 height total energy of satellite is equal to potential energy of satellite at 2R .

∴ option (a) is correct