# A-level Physics (Advancing Physics)/Gravitational Potential/Worked Solutions

G = 6.67 x 10-11 m3kg-1s-2

g = 9.81 ms-2

1. What is the gravitational potential at the Earth's surface? (mass of Earth = 5.97 x 1024 kg,radius of Earth = 6371 km)

$V_{grav} = \frac{-GM}{r} = \frac{-6.67 \times 10^{-11} \times 5.97 \times 10^{24}{ MJkg}^{-1}}{6371000} = 62.5\mbox{ MJkg}^{-1}$

2. Taking the Earth's surface as Vgrav = 0, what is the gravitational potential 2m above the ground?

$V_{grav} \approx g\Delta h = 9.81 \times 2 = 19.62\mbox{ Jkg}^{-1}$

3. A 0.2kg firework reaches a gravitational potential relative to the ground of 500Jkg-1. If the firework is 30% efficient, how much energy was expended to get there?

$V_{grav} = \frac{E_{grav}}{m}$

$E_{grav} = mV_{grav} = 0.2 \times 500 = 100\mbox{ J}$

However, this is only 30% of the energy expended, so:

$E_{expended} = \frac{100}{0.3} \approx 333\mbox{ J}$

4. Express gravitational potential in terms of gravitational force.

$V_{grav} = \frac{\int{F_{grav}}\; dr}{m}$

5. Draw the equipotentials and field lines surrounding the Earth.