IB Physics SL D.1.1 Kepler’s three laws of orbital motion

Apply Kepler’s laws to orbital data and diagrams, linking period, radius and changing speed in elliptical motion.

Exam points

use Kepler’s third law, T^2 ∝ r^3, to calculate an unknown orbital radius or period from given data
identify the correct law statement from a diagram: ellipse with the star at a focus, or equal areas in equal times
explain from Kepler’s second law that speed or kinetic energy is greater when the planet is nearer the star

[Maximum number: 3]

A satellite powered by solar cells directed towards the Sun is in a polar orbit about the Earth.

Figure for Question D.1.1—Kepler’s three laws of orbital motion question 1 — IB Physics SL

The satellite is orbiting the Earth at a distance of 6600 km from the centre of the Earth.

Determine the orbital period for the satellite.

Mass of Earth $=6.0 \times 10^{24} \mathrm{~kg}$

$\begin{aligned} & \frac{m v^{2}}{r}=G \frac{M m}{r^{2}} & \text { leading to } T^{2}=\frac{4 \pi^{2} r^{3}}{G M} & T=5320 \text { «s» } \end{aligned}$

Alternative 2
$\begin{aligned} & \text { « } V=\sqrt{\frac{G m_{E}}{r}} \text { » }=\sqrt{\frac{6.67 \times 10^{-11} \times 6.0 \times 10^{24}}{6600 \times 10^{3}}} \text { OR } 7800 \text { « } \mathrm{ms}^{-1} \text { » } & \text { distance }=2 \pi r=2 \pi \times 6600 \times 10^{3} \text { « } \mathrm{m} \text { » or } 4.15 \times 10^{7} \text { « } \mathrm{m} \text { » } & \text { « } T=\frac{d}{v}=\frac{4.15 \times 10^{7}}{7800} \text { » }=5300 \text { « } \mathrm{S} \text { » } \end{aligned}$

Accept use of $\omega$ nistead of v

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D.1.1—Kepler’s three laws of orbital motion question 1