Problem 20 - A Complete Resource Book in Physics for JEE Main

Question

A planet is revolving around the sun in elliptical orbit. Its closest distance from the sun is \(r\) and the farthest distance is \(R\). If the orbital velocity of the planet closest to the sun be \(v\), then what is the velocity at the farthest point? (A) \(\overline{v r} / R\) (B) \(v R / r\) (C) \(v\left(\frac{r}{R}\right)^{1 / 2}\) (D) \(v\left(\frac{R}{r}\right)^{1 / 2}\)

Step-by-Step Solution

Answer

The short answer is: \(vf = \frac{\overline{v r}}{R}\)

Step 1: Angular Momentum at Closest Point (Perihelion)

The angular momentum at the closest point to the sun can be given by the formula (Leaving out the constant mass, as it will cancel in the end): L1 = r * v

Step 2: Angular Momentum at Farthest Point (Aphelion)

Now, we need to find the angular momentum at the farthest point from the sun. Let's call the velocity at the farthest point vf. L2 = R * vf

Step 3: Apply the Conservation of Angular Momentum

Since there is no external torque acting on the planet, the conservation of angular momentum applies. This means that L1 = L2. r * v = R * vf

Step 4: Solve for the Velocity at the Farthest Point (vf)

Now, we can solve for vf in the equation from the previous step: vf = (r * v) / R This matches option (A) in the exercise. Therefore, the velocity of the planet at its farthest point from the sun is: \(vf = \frac{\overline{v r}}{R}\)

Key Concepts

Elliptical Orbit

In our solar system, planets revolve around the sun in elliptical orbits. Unlike a perfect circle, an ellipse is an oval shape. This means that there are points in the orbit where the planet is closer to the sun, known as the perihelion, and points where it's farther away, called the aphelion.

Key features of an elliptical orbit include:

• The two focal points, with the sun occupying one of them.
• Varying distances from the sun at different points in the orbit.
• The major and minor axes, defining the longest and shortest diameters of the ellipse.

Because the distance between the planet and the sun changes, the gravitational force and speed of the planet vary throughout the orbit. It's this unique elliptical shape that causes the planet to move faster when closer to the sun and slower when further away.

Conservation of Angular Momentum

One of the key principles influencing planetary motion is the conservation of angular momentum. This principle states that if no external forces act on a system, the total angular momentum remains constant.

In the context of a planet orbiting the sun, this means:

• At the perihelion, the planet moves faster because it's closer to the sun.
• At the aphelion, the planet moves slower because it is further from the sun.

Angular momentum can be expressed as the product of the radius of the orbit and the velocity of the planet at that point. So, for a planet: \[ L = r imes v \]Where:

• \(L\) is angular momentum.
• \(r\) is the radius (distance to the sun).
• \(v\) is the velocity.

By applying this concept, we find that the planet speeds up and slows down to maintain constant angular momentum as it travels along its elliptical path.

Orbital Velocity

Orbital velocity is the speed at which a planet moves along its orbit. It's determined by the gravitational pull of the sun and the shape of the orbit.

Important aspects of orbital velocity include:

• It is highest at the perihelion (closest point to the sun).
• It is lowest at the aphelion (farthest point from the sun).

By using the conservation of angular momentum, the velocity at the aphelion can be calculated. For a planet with velocity \(v\) at the perihelion, and respective distances \(r\) and \(R\), the velocity at the aphelion \(v_f\) can be expressed as:\[ v_f = \frac{v imes r}{R} \]This means that the further a planet is from the sun, the slower it needs to move to keep its orbit stable. Understanding these velocities helps explain why planets have different speeds based on their position in the orbit.