For a satellite in a circular orbit at radius r
Gravitational Fields - OCR A-Level Physics
- For a satellite in a circular orbit at radius r:
- Kinetic energyThe capacity to do work. Measured in joules (J).The energyThe capacity to do work. Measured in joules (J). an object possesses due to its motion.: $KE = \frac{1}{2}mv^2 = \frac{GMm}{2r}$ (always positive).
- Potential energyThe capacity to do work. Measured in joules (J).: $PE = -\frac{GMm}{r}$ (always negative).
- Total energy: $E = KE + PE = -\frac{GMm}{2r}$ (always negative for a bound orbit).
- The total energy is negative, meaning the satellite is bound to the central body.
- Notice: $|PE| = 2 \times KE$, and $E = -KE = PE/2$.
- To move a satellite to a higher orbit, you must add energy. This increases both KE at the point of boost AND the orbital PE. The total energy becomes less negative (closer to zero).
- If total energy reaches zero, the satellite has exactly escape velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The minimum speed at which an object must be launched from the surface of a body to escape its gravitational field entirely (reach infinity with zero kinetic energyThe energy an object possesses due to its motion.). and follows a parabolic path.
Worked Example
A 500 kg satellite orbits Earth at a height of 400 km. Calculate its total energy. ($M_E = 5.97 \times 10^{24} \text{ kg}$, $R_E = 6.37 \times 10^{6} \text{ m}$)
Show Solution
1
$r = R_E + h = 6.37 \times 10^6 + 400 \times 10^3 = 6.77 \times 10^6 \text{ m}$.
2
$E = -GMm/(2r).$
3
$E = -(6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 500) / (2 \times 6.77 \times 10^6).$
4
$E = -(1.99 \times 10^{17}) / (1.354 \times 10^7).$
5
$E = -1.47 \times 10^{10} \text{ J} \approx -14.7 \text{ GJ}$.
Answer
$$E \approx -14.7 \text{ GJ}$$
Common Mistake
MEDIUM
Wrong: Expecting a satellite in a higher orbit to have more kinetic energyThe energy an object possesses due to its motion..
Right: A satellite in a higher orbit has LESS kinetic energy (slower speed) but more (less negative) total energy. The energy input goes primarily into increasing GPE.
Right: A satellite in a higher orbit has LESS kinetic energy (slower speed) but more (less negative) total energy. The energy input goes primarily into increasing GPE.