If the potential energy of a body on a planet is numerically U and the escape velocity for the same body be ve for the same planet then U / ve will be :

Question

If the potential energy of a body on a planet is numerically U and the escape velocity for the same body be ve for the same planet then U / ve will be : (A) (U/ve)=m √(G M/2 R) (B)  (U/ve)=m√(GM/2R)  (C)  (U/ve)=m√(2GM/R)  (D)  (U/ve)=m(GM/R)

Tardigrade Admin · Accepted Answer

The work obtained in bringing a body from infinity to a point in a gravitational field is called the gravitational potential energy of the body at that point.

U = - \frac{G M _{m}}{R} ... (1)

where

G

is gravitational constant,

M

is mass of earth

R

its radius and

m

is mass of body.
Also for a body projected upwards at a certain velocity of projection the body will go out of the gravitational field of the earth and will never return to the earth, this initial velocity is called escape velocity.

v_{e} = \frac{2 GM}{R} ... (2)

Since, work is required to take a body from earth's surface to infinity, we have

U = + \frac{G M _{m}}{R} .... (3)

Dividing Eq. (3) by (2), we get

\frac{U}{v _{e}} = m \frac{GM}{2 R}

Q. If the potential energy of a body on a planet is numerically $U$ and the escape velocity for the same body be $v_{e}$ for the same planet then $U / v_{e}$ will be :

$\frac{U}{v _{e}} = m \frac{GM}{2 R}$

$\frac{U}{v _{e}} = m \frac{GM}{2 R}$

$\frac{U}{v _{e}} = m \frac{2 GM}{R}$

$\frac{U}{v _{e}} = m \frac{GM}{R}$

Q. If the potential energy of a body on a planet is numerically U and the escape velocity for the same body be ve​ for the same planet then U/ve​ will be :

ve​U​=m2RGM​​

ve​U​=m2RGM​​

ve​U​=mR2GM​​

ve​U​=mRGM​

Q. If the potential energy of a body on a planet is numerically $U$ and the escape velocity for the same body be $v_{e}$ for the same planet then $U / v_{e}$ will be :

$\frac{U}{v _{e}} = m \frac{GM}{2 R}$

$\frac{U}{v _{e}} = m \frac{GM}{2 R}$

$\frac{U}{v _{e}} = m \frac{2 GM}{R}$

$\frac{U}{v _{e}} = m \frac{GM}{R}$