Kepler's laws and orbital motion
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Question
State Kepler's third law.
Answer
The **square** of a planet's orbital period is **proportional** to the **cube** of its orbital radius: $T^{2} \propto r^{3}$.
Question
State Kepler's first law.
Answer
Each planet moves in an **ellipse** with the Sun at one **focus** of the ellipse.
Question
State Kepler's second law.
Answer
A planet moves **faster when nearer the Sun** and **slower when farther away** (it sweeps out equal areas in equal times).
Question
Full form of Kepler's third law for a circular orbit?
Answer
$T^{2} = \dfrac{4\pi^{2}r^{3}}{GM}$ — derived from $g = GM/r^{2}$ and $a = 4\pi^{2}r/T^{2}$. M is the mass being orbited.
Question
How do you compare two orbits round the same body without knowing G or M?
Answer
Use $\dfrac{T_{A}^{2}}{r_{A}^{3}} = \dfrac{T_{B}^{2}}{r_{B}^{3}}$ — the constant $4\pi^{2}/(GM)$ cancels.
Question
What shape is a graph of T² against r³?
Answer
A **straight line through the origin** — because $T^{2}/r^{3}$ is a constant.
Question
If a planet's orbit radius is 4× larger, how much longer is its period?
Answer
$(T_{B}/T_{A})^{2} = 4^{3} = 64$, so $T_{B}/T_{A} = \sqrt{64} = 8$ — **8 times** longer.
Question
Why does a planet's kinetic energy change over its elliptical orbit?
Answer
By the second law it moves **faster when closer** to the Sun (more kinetic energy) and **slower when farther** (less), so its speed and kinetic energy vary.
Question
What keeps a planet in orbit?
Answer
The **gravitational pull of the Sun**, directed inward (centripetal), which bends the planet's path into a closed orbit.
Question
In Kepler's third law, watch the powers — which is which?
Answer
T is **squared**, r is **cubed**: $T^{2} \propto r^{3}$. Mixing them up is the classic mistake.
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Gravitational fields
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