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Question: The minimum and maximum distances of a planet revolving around the sun are \(\mathrm{r}\) and R. If ...

The minimum and maximum distances of a planet revolving around the sun are r\mathrm{r} and R. If the minimum speed of planet on its trajectory is v0,\mathrm{v}_{0}, then its maximum speed will be:
A.v0Rr\dfrac{{{\text{v}}_{0}}\text{R}}{\text{r}}
B.v0rR\dfrac{{{\text{v}}_{0}}\text{r}}{R}
C.v0R2r2\dfrac{{{\text{v}}_{0}}{{\text{R}}^{2}}}{{{\text{r}}^{2}}}
D.v0r2R2\dfrac{{{\text{v}}_{0}}{{\text{r}}^{2}}}{{{R}^{2}}}

Explanation

Solution

The planet's orbital radius and angular velocity in the elliptical orbit will vary. The centre of the Sun is always located at one focal point of the orbital ellipse. The Sun has only one focus. The planet follows the ellipse in its orbit, meaning that as the planet goes around its orbit, the distance from the planet to the Sun is constantly changing. Calculate Velocity by using Kepler law Since the motion is orbital There will be angular momentum and By using Kepler 2nd law we can find the velocity.

Formula used:
mv0R=mv′r\mathrm{mv}_{0} \mathrm{R}=\mathrm{mv}^{\prime} \mathrm{r}

Complete step by step solution:
The second law of Kepler states that a planet's real velocity, with the sun taken as the origin, is constant. He proved in 1684, with the aid of his laws of motion, that any planet attracted to a fixed center sweeps out equal areas at equal intervals of time.
By Kepler's law, the areal velocity =L2 m=\dfrac{\mathrm{L}}{2 \mathrm{~m}} is a constant and hence L\mathrm{L} (angular momentum) is
constant.
mvr=\mathrm{mvr}= constant (at any point of trajectory)
Minimum speed (v0)\left(\mathrm{v}_{0}\right) is possible at the maximum distance (R) from sun and maximum
speed (v′)\left(\mathrm{v}^{\prime}\right) is possible at the minimum distance (r) from the sun.
mv0R=mv′r\mathrm{mv}_{0} \mathrm{R}=\mathrm{mv}^{\prime} \mathrm{r}
∴v′=v0Rr\therefore {{\text{v}}^{\prime }}=\dfrac{{{\text{v}}_{0}}\text{R}}{\text{r}}
its maximum speed will be: v0Rr\dfrac{{{\text{v}}_{0}}\text{R}}{\text{r}}

Hence, the correct option is (A).

Note:
Kepler's second law states that at equal times, a planet sweeps out equal areas, i.e. the area divided by time, called the velocity of the planet, is constant. The direction of the areal velocity is the same as that of, i.e. perpendicular to the planet containing r and a and is directed as provided by the rule of the right hand. An ellipse is defined as the set of all points, such that a constant is the sum of the distance from each point to two focal points