Question

Choose the correct alternative:
(a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic / potential energy.
(b) The energy required to launch an orbiting satellite out of earths gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earths influence.

Answer 1

Let $\text{Height of satellite+Radius of earth=R}$

(a)

For a satellite,

$\text{Gravitational Force = Centripetal Force}$

$GMm/R^2=m{v}^2/R..........\left(1\right)$

$U=-GMm/R$

$K.E.=\frac{1}{2}m{v}^2$

$T.E.=U+K.E.$

$T.E.=-GMm/R+\frac{1}{2}m{v}^2............\left(2\right)$

From (1) and (2),

$T.E.=-\frac{1}{2}m{v}^2=-K.E.$

Hence, if the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic energy.

(b)

For an object to escape, the energy required should be such that total energy is zero.

For a satellite, $T.E.=-\frac{GMm}{2R}$ from part (a)

Extra energy for satellite, $E_1=\frac{GMm}{2R}...........\left(1\right)$

For a stationary object, $T.E.=U=-\frac{GMm}{R}$

Extra energy for object, $E_2=\frac{GMm}{R}...........\left(2\right)$

From (1) and (2),

$\frac{E_1}{E_2}=\frac{1}{2}$

Hence, the energy required to launch an orbiting satellite out of earth's gravitational influence is less than the energy required to project a stationary object at the same height (as the satellite) out of earth's influence.