Question

A satellite orbits the earth at a height of 400km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 × 1024 kg; radius of the earth = 6.4 × 106 m; G = 6.67 × 10-11 Nm2kg-2.

Answer 1

HintApplying concept of energy conservation i.e. calculating total energy (TE) which is equal to the sum of Kinetic Energy (KE) and Potential Energy (PE). In the given question KE of satellite is with which it is rotating 400km above around the Earth and PE is gravitational energy between the Earth and the satellite. Our aim is to calculate these quantities and then we are done.

 Complete step-by-step solution:First we have to calculate the KE of the satellite
\[KE = \dfrac{1}{2}m{v^2}\]
Here we know m but not v, for that we are equalizing the Gravitational Force between the Earth and satellite with the Centripetal force experienced by the satellite.
Gravitational Force formula \[F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}\] ---(1)
Centripetal Force formula \[{F_c} = \dfrac{{m{v^2}}}{r}\] ---(2)
According to question:
r = r + R ( 400km + Radius of Earth )
${m_1}$ =M ( mass of Earth )
${m_2}$ = m (mass of satellite )
Equating equation (1) and (2)
\[\dfrac{{m{v^2}}}{r} = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}\]
\[\dfrac{{m{v^2}}}{{r + {R^{}}}} = \dfrac{{G.M.m}}{{{{(r + R)}^2}}}\]
\[{v^2} = \dfrac{{G.M.}}{{(r + R)}}\]
Not putting value of v2 in KE equation
\[KE = \dfrac{1}{2}m\dfrac{{G.M.}}{{(r + R)}}\]
\[KE = \dfrac{1}{2} \times 200 \times \dfrac{{6.67{\text{ }} \times {\text{ }}{{10}^{ - 11}} \times 6.0 \times {{10}^{24}}}}{{(0.4 + 6.4 \times {\text{ }}{{10}^6})}}\]
\[KE = 5.89 \times {10^9}J\]
Now PE of the satellite above earth height r is \[PE = - \dfrac{{GMm}}{{R + r}}\]
\[PE = - \dfrac{{6.67{\text{ }} \times {\text{ }}{{10}^{ - 11}} \times 6.0 \times {{10}^{24}} \times 200}}{{(0.4 + 6.4) \times {\text{ }}{{10}^6}}}\]
\[PE = - 11.78 \times {10^9}J\]
\[Now{\text{ }}TE = {\text{ }}KE{\text{ }} + {\text{ }}PE{\text{ }} = \;\]\[5.89 \times {10^9}\]\[ - 11.78 \times {10^9}J\]
\[TE = 5.89 \times {10^9}J\]
This is the energy required for satellites so that it moves out of Earth’s gravitational influence.

Note: Whenever we calculate such a question try to visualize what is happening; what questioners want to ask. Create a map of the question in the mind and then what will be the processes you will do step by step. In the given question PE is –ve as we know PE is energy required to bring an object from infinity to that point .And again by making PE zero it means we are moving satellites out the Gravitational field of Earth.