Question

A planet moves around the sun. At a given point P, it is closest from the sun at a distance \[{{d}_{1}}\] and has a speed \[{{v}_{1}}\]. At another point Q, when it is farthest from the sun at a distance \[{{d}_{2}}\], its speed will be:
\[A.\,\dfrac{d_{1}^{2}{{v}_{1}}}{{{d}_{2}}}\]
\[B.\,\dfrac{{{d}_{2}}{{v}_{1}}}{{{d}_{2}}}\]
\[C.\,\dfrac{{{d}_{1}}{{v}_{1}}}{{{d}_{2}}}\]
\[D.\,\dfrac{d_{2}^{2}{{v}_{1}}}{d_{1}^{2}}\]

Answer 1

Hint: This question is based on the concept of the law of conservation of the angular momentum, that is, the angular momentum of the planets revolving around the sun in an elliptical orbit remains constant. Using this formula we will find the velocity of the planet at the point Q.

Complete step by step answer:
According to the law of conservation of the angular momentum, the planets revolving around the sun in an elliptical orbit have the angular momentum to be constant.
\[\dfrac{L}{2m}=\text{constant}\]
\[\Rightarrow \dfrac{mvr}{2m}=\text{constant}\]
Where m is the mass of the planet, r is the distance between the sun and the planet and v is the speed of the planet (linear speed)
From given, we have the data, At a given point P, the planet is closest from the sun at a distance \[{{d}_{1}}\], and has a speed \[{{v}_{1}}\]. At another point Q, the planet is farthest from the sun at a distance\[{{d}_{2}}\].
Using the law of conservation of the angular momentum equation, we get,
\[\begin{align}
  & m{{v}_{1}}{{d}_{1}}=m{{v}_{2}}{{d}_{2}} \\
 & \therefore {{v}_{1}}{{d}_{1}}={{v}_{2}}{{d}_{2}} \\
\end{align}\]
The common term, that is, the mass m cancels out.
Rearrange the terms of the above equation to obtain an expression in terms of the second velocity. So, we have,
\[{{v}_{2}}=\dfrac{{{v}_{1}}{{d}_{1}}}{{{d}_{2}}}\]
Where distance \[{{d}_{1}}\], and has a speed \[{{v}_{1}}\] at a point P from where the planet is closest to the sun and distance \[{{d}_{2}}\] at a point Q from where the planet is farthest from the sun.

As the equation of the velocity at a point Q from where the planet is farthest from the sun at a distance of \[{{d}_{2}}\] is obtained to be equal to \[\dfrac{{{v}_{1}}{{d}_{1}}}{{{d}_{2}}}\], thus, the option (C) is correct.

Note:
The concept, that is, the angular momentum of the planets revolving around the sun in an elliptical orbit remains constant should be known. Or, simply, by using the law of conservation of the angular momentum also we can derive the expression for the velocity of the planet.