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Universal Law of Gravitation

A planet movi...

Question

A planet moving along an elliptical orbit is closest to the sun at a distance $r_{1}$ and farthest away at a distance of $r_{2}$ . If $v_{1}$ and $v_{2}$ are the linear velocities at these points respectively, then the ratio $\frac{v_{1}}{v_{2}}$ is

\frac{r_{1}}{r_{2}}

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{(\frac{r_{1}}{r_{2}})}^{2}

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\frac{r_{2}}{r_{1}}

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{(\frac{r_{2}}{r_{1}})}^{2}

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Solution

The correct option is C $\frac{r_{2}}{r_{1}}$
Given that $v_{1}$ is the speed of the planet when it is at a distance $r_{1}$ and $v_{2}$ is the speed when it is at a distance $r_{2}$ as shown in the figure.

Since the torque on the planet about the sun is zero, its angular momentum will be conserved.

Now using conservation of angular momentum at $P$ and $A$ ,

$m v_{1} r_{1} = m v_{2} r_{2}$

$\Rightarrow \frac{v_{1}}{v_{2}} = \frac{r_{2}}{r_{1}}$

Hence, option $(c)$ is correct.

Why this question: To help students understand the application of conservation of angular momentum in planetary motion.

Key Concept: The angular momentum of a planet is conserved about the star as the force passes through the star itself and hence external torque is zero.

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Universal Law of Gravitation

Standard X Physics

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A planet moving along an elliptical orbit is closest to the sun at a distance r1 and farthest away at a distance of r2. If v1 and v2 are the linear velocities at these points respectively, then the ratio v1v2 is

A planet moving along an elliptical orbit is closest to the sun at a distance $r_{1}$ and farthest away at a distance of $r_{2}$ . If $v_{1}$ and $v_{2}$ are the linear velocities at these points respectively, then the ratio $\frac{v_{1}}{v_{2}}$ is