Question

Find out energy required (in GigaJoule) to escape a space shuttle of 1000kg mass from surface to earth. ( ${R_e} = 6400$ km):

A) $20$
B) $64$
C) $50$
D) $28$

Answer 1

Firstly, we can do the conversion of the Gigajoules to joules by $1GJ\, = \,{10^9}J$ . Since the velocity of the space shuttle is not given therefore we will use the concept of conservation of Energy. The energy of the system will be conserved. Also in order to solve this question, we must know the value of the Universal Gravitational constant.

Complete step by step solution:
Now we know that the energy of any system is conserved. We also know that velocity is not given. So, we cannot use any equation for kinetic energy. Therefore, the potential energy is the only energy the space shuttle is having in the initial stage.
Now, the Potential energy is the energy that an object has by virtue of its height. So, the space shuttle is on the surface of Earth and the force acting on the space shuttle is the gravitational force of the Earth. It is assumed that the gravitational force of Earth acts from the center and the surface of Earth is at a height of $6400$ km.
Therefore, the potential Energy of the space shuttle at the initial stage will be given by the formula:
$PE = \, - \dfrac{{GmM}}{R}$
Where, $PE$ denotes the potential Energy, generally potential energy is denoted by the symbol U or V
$G$ denotes the universal gravitational constant
$m$ is the mass of the space shuttle
$M$ is the mass of Earth.
$R$ is the distance from the center of Earth.
The negative sign in the formula indicated that the object is bound to Earth and it will move in the opposite direction. Potential energy is always taken as negative.
Let the energy required for the space shuttle to escape be E
Then, from energy conservation we have
$\, - \dfrac{{GmM}}{R} + E = 0$
$\, \Rightarrow E = \dfrac{{GmM}}{R}$

As value of mass of Earth and Gravitational constant is not given
Therefore, we will use:
$\dfrac{{GmM}}{R} = g$
Where, $g$ is the acceleration due to gravity.
Now, the above equation implies:
$\, \Rightarrow E = m\left( {\dfrac{{GM}}{{{R^2}}}} \right)$

$\, \Rightarrow E = m\left( g \right)R$

Now, we will substitute the values from the given question: $g = 10\,m/{s^2}$
$\, \Rightarrow E = 1000 \times 10 \times 6.4 \times {10^6}$
$\, \Rightarrow E = 64 \times 1{0^9}$ Joules

That is, $64$ GigaJoules.

Hence, option B is the correct option.

Note: The relation between Giga joules and Joules should be known in order to solve this question. Also the value of acceleration is taken as $g = 10\,m/{s^2}$ to simplify the calculation. Be careful when using decimal conversions and power conversions, mistakes are likely to happen.