First slide

Gravitaional potential energy

Question

A spaceship is launched into a circular orbit of radius R close to the surface of the earth. The additional velocity to be imparted to the spaceship in the orbit to overcome the earth’s gravitational pull is (g → acceleration due to gravity)

Moderate

A

$\large 1.414Rg$
B

$\large 1.414\sqrt {gR}$
C

$\large 0.414 {Rg}$
D

$\large 0.414 \sqrt {gR}$

Solution

$\large {V_{additional}} = {V_e} - {V_0} = \sqrt 2 {V_0} - {V_0} = {V_0}\left( {\sqrt 2 - 1} \right)$
$\large {V_{additional}} = 0.414\,{V_0} = 0.414\,\sqrt {gR}$
$\large V_e=\sqrt {\frac {2GM}{R}}=$ Escape velocity at a radial distance R from the cenre of earth.
V₀ = Orbital speed = $\large \sqrt \frac {GM}{R}=\sqrt {gR}$
$\large \therefore$ Additional velocity = V_e - V₀⇒ (√2 - 1)V₀ = $\large 0.414\sqrt {gR}$

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