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Dimensional Analysis

Let the speed...

Question

Let the speed of the planet at the perihelion P in Figure. be $v_{P}$ and the Sun-planet distance SP be $r_{P}$ . Relate $r_{P}, v_{P}$ to the corresponding qualities at the aphelion $(r_{A}, v_{A})$ . Will the planet take equal times to traverse BAC and CPB?

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Solution

Referring Figure, we note that $_{P}$ and $_{P}$ are perpendicular to each other. Similarly, $_{A}$ and $_{A}$ are perpendicular to each other.
Using the law of conservation of angular momentum. Angular momentum of the planet at P $=$ Angular momentum of the planet at A
$\Rightarrow m_{P} v_{P} r_{P} = m_{A} v_{A} r_{A}$
or $\frac{v_{P}}{v_{A}} = \frac{r_{A}}{r_{P}}$
Since $r_{A} > r_{P}, v_{P} > v_{A}$
Here area SBAC is greater than area SCPB. As the areal velocity of a planet is constant around the Sun, i.e., equal areas are swept in equal times. Hence the planet will take longer time to traverse BAC than CPB.

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