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Elliptical Orbits and Kepler's Laws

The following...

Question

The following figure shows the elliptical path of a planet around the sun at point $O$ . $A O B$ and $C O D$ have equal areas. Suppose $t_{1}$ and $t_{2}$ be the time taken by the planet to go from $A$ to $B$ and from $D$ to $C$ respectively. Choose the correct option.

t_{1} < t_{2}

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t_{1} > t_{2}

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t_{1} = t_{2}

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t_{1} \neq t_{2}

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Solution

The correct option is C $t_{1} = t_{2}$
According to Kepler’s second law, the line joining a planet and the Sun sweeps equal areas in equal intervals of time.
Since the line joining the planet to the sun sweeps out equal areas in travelling from $A$ to $B$ and from $D$ to $C$ , it would have taken same time in both the cases.
Hence, $t_{1} = t_{2}$ .

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