The correct answer is B because:

The total energy given to the space probe E contributes to its kinetic energy and to the change in its gravitational potential energy, $∆ E_{P}$ :
- E = $E_{K} + ∆ E_{P}$
- $\frac{3 G M_{m}}{4 R_{E}}$ $= E_{K} + ∆ E_{P}$
The space probe’s kinetic energy $E_{K} = \frac{1}{2} m v^{2} = \frac{1}{2} m {(\sqrt{\frac{G M}{R}})}^{2} = \frac{G M m}{2 R}$
- Where R is the height of the space probe above the Earth’s surface
The space probe’s change in gravitational energy, as it launches from a radial distance $R_{E}$ to a height R, is given by:
- $∆ E_{P}$ = $- \frac{G M m}{∆ r}$ = –GMm $(\frac{1}{R} - \frac{1}{R_{E}})$ = $\frac{G M m}{R_{E}} - \frac{G M m}{R}$
Therefore:
- $\frac{3 G M m}{4 R_{E}} = E_{k} + ∆ E_{P}$
- $\frac{3 G M m}{4 R_{E}} = \frac{G M m}{2 R} + \frac{G M m}{R_{E}} - \frac{G M m}{R}$
- $\frac{3 G M m}{4 R_{E}} - \frac{G M m}{R_{E}} = \frac{G M m}{2 R} - \frac{G M m}{R}$
- $\frac{3 G M m}{4 R_{E}} - \frac{4 G M m}{4 R_{E}} = \frac{G M m}{2 R} - \frac{2 G M m}{2 R}$
- $- \frac{G M m}{4 R_{E}} = - \frac{G M m}{2 R}$
- $- \frac{1}{4 R_{E}} = - \frac{1}{2 R}$
- 2R = 4 $R_{E}$
- R = 2 $R_{E}$
Therefore, the height of the space probe R = 2 $R_{E}$

This is a really good question to practise your understanding of how energy is transferred to objects in orbit.

The basis for this question is the total energy is transferred to moving the space probe (i.e., its kinetic energy) and to changing its gravitational potential energy (i.e., it has some GPE at the Earth’s surface, and it will have some different GPE higher up). From here, it’s a case of finding the correct mathematical expression for each piece of the puzzle: the kinetic energy $E_{K}$ and the change in GPE, $∆ E_{P}$ .

Since the kinetic energy $E_{K}$ ∝ v², you need an expression for the velocity of an object of mass m in orbit. This velocity v = $\sqrt{\frac{G M}{r}}$ , where M is the mass of the body being orbited and r is the distance between the two bodies. You should be comfortable with deriving this equation – for your reference, this is written out below:

7-3-mcq-q2-ma-hard-aqa-a-level-physics

Orbits of Planets & Satellites (AQA A Level Physics)

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