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Q.

In planetary motion the areal velocity of position vector of a planet depends on angular velocity ( $ω$ ) and the distance of the planet from sun (r). If so, the correct relation for areal velocity is

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a

\frac{dA}{dt} \propto \sqrt{ω r}

b

\frac{dA}{dt} \propto ω r

c

\frac{dA}{dt} \propto ω^{2} r

d

\frac{dA}{dt} \propto ω r^{2}

answer is C.

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Detailed Solution

\frac{dA}{dt} = \frac{d}{dt} (\frac{1}{2} \times OA \times AB) = \frac{d}{dt} (\frac{1}{2} \times r \times r \times θ)

Question Image

$∴ AB = r θ$

$= \frac{d}{dt} \times \frac{r^{2} θ}{2} = \frac{1}{2} r^{2} \frac{d θ}{dt} = \frac{1}{2} r^{2} ω$

$\frac{dA}{dt} \propto r^{2} ω$

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