A planet revolving in elliptical orbit has
I. a constant velocity of revolution
II. has the least ve

Q: Why does the planet's velocity change in an elliptical orbit?

As the planet moves closer to the Sun, the gravitational force does positive work, increasing its kinetic energy and speed. Conversely, as it moves away, the force does negative work, slowing it down.

Q: What exactly is 'areal velocity'?

Areal velocity is the rate at which the imaginary line connecting the planet to the Sun sweeps out area over time. Mathematically, it is $dA/dt$.

Q: Is angular momentum conserved in this motion?

Yes! Since the gravitational force is a central force (always directed towards the Sun), the torque is zero. Therefore, the angular momentum of the planet remains constant.

Q: How is areal velocity related to angular momentum?

Areal velocity is directly proportional to angular momentum. Specifically, $dA/dt = L / (2m)$, where $L$ is the angular momentum and $m$ is the mass of the planet. Since $L$ is constant, areal velocity is also constant.

Q: Does Kepler's Second Law apply to circular orbits too?

Absolutely. A circular orbit is just a special case of an elliptical orbit where the two foci coincide. In a circular orbit, both the linear speed and the areal velocity remain constant.

Question

A planet revolving in elliptical orbit has
I. a constant velocity of revolution
II. has the least velocity when it is nearest to the Sun
III. its areal velocity is directly proportional to its velocity
IV. areal velocity is inversely proportional to its velocity.
V. to follow a trajectory such that the areal velocity is constant.
Choose the correct answer from the options given below.

Options:

(A) Only I

(B) Only IV

(C) Only III

(D) Only V

Sigma Percentile · Accepted Answer

### The Elliptical Journey

Imagine you are observing a planet as it journeys through the cosmos. According to Kepler's First Law, this journey is not a perfect circle, but an **elliptical orbit**, with the Sun positioned not at the center, but at one of the focal points. This fundamental geometry sets the stage for some fascinating dynamic behavior. Because the Sun is off-center, the planet's distance from the Sun is constantly changing as it travels along its path.

### The Dance of Velocity

Does the planet cruise at a leisurely, constant speed? Absolutely not! As the planet approaches the Sun (a point known as **perihelion**), the Sun's gravitational pull intensifies. This force does positive work on the planet, accelerating it to its maximum velocity, $v_{max}$.

Conversely, as the planet swings away towards the farthest point in its orbit (**aphelion**), it has to fight against the Sun's gravity. The gravitational force does negative work, causing the planet to slow down to its minimum velocity, $v_{min}$. Therefore, statement I (constant velocity) and statement II (least velocity when nearest) are completely incorrect.

### Kepler's Masterpiece: The Law of Areas

This brings us to the heart of the problem: **Kepler's Second Law of Planetary Motion**, beautifully known as the Law of Areas. Kepler discovered that if you draw an imaginary line from the center of the Sun to the center of the planet, this line will sweep out **equal areas in equal intervals of time**.

Mathematically, this means the rate of change of area with respect to time, known as the **areal velocity** ($\frac{dA}{dt}$), is a universal constant for that orbit. This elegance arises directly from the conservation of angular momentum. Since gravity is a central force, it exerts no torque, keeping the angular momentum $L$ constant. The areal velocity is simply $\frac{L}{2m}$, proving it must remain constant.

### Analyzing the Options

Armed with this knowledge, let's evaluate the remaining statements. Statements III and IV suggest that areal velocity changes proportionally or inversely with linear velocity. We know this is false because areal velocity is strictly constant.

Finally, we look at statement V: "to follow a trajectory such that the areal velocity is constant." This is the exact definition of Kepler's Second Law! The planet naturally follows this trajectory, making statement V the only correct assertion. Therefore, the correct option is **(d)**.

Select Answer:

Visualized Solution

$E l l i pt i c a l O r bi t$

$V a r ia t i o n o f V e l oc i t y$

$K e pl e r^{'} s S eco n d L a w$

$C o n s t an t A r e a l V e l oc i t y$

The Sigma Insight: Kepler's Laws of Planetary Motion

The Elliptical Journey

The Dance of Velocity

Kepler's Masterpiece

Analyzing the Options

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