A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very s

Q: Why is the Earth located at the focus of the ellipse rather than the center?

According to Kepler's First Law of Planetary Motion, all planets/satellites move in elliptical orbits with the central massive body (like the Sun or Earth) situated at one of the two foci. This is a direct mathematical consequence of the inverse-square nature of the gravitational force.

Q: If the acceleration is always directed towards the Earth, why doesn't the satellite fall into it?

The satellite has a high tangential velocity. While gravity constantly pulls it inward, changing its direction of motion, the satellite's forward inertia keeps it moving along the curved orbital path, essentially 'falling around' the Earth continuously.

Q: Why is the angular momentum vector constant in direction as well as magnitude?

Because the net torque on the satellite is exactly zero. Since torque is the rate of change of angular momentum ($\vec{\tau} = d\vec{L}/dt$), a zero torque means the vector $\vec{L}$ cannot change at all, preserving both its length (magnitude) and its orientation in 3D space.

Q: How do kinetic and potential energy vary in an elliptical orbit?

As the satellite moves closer to the Earth (decreasing $r$), its gravitational potential energy $U = -GMm/r$ becomes more negative (decreases), while its kinetic energy $K$ increases (it speeds up). The opposite happens as it moves farther away. Their sum, $E = K + U$, remains strictly constant.

Q: Is the linear momentum constant at any point in the orbit?

No. Linear momentum $\vec{p} = m\vec{v}$ is a vector. Since the direction of motion is constantly changing along the ellipse, the direction of $\vec{p}$ is never constant. Furthermore, its magnitude $p = mv$ also changes continuously because the speed $v$ varies throughout the orbit.

Question

A satellite $S$ is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth. Then,

Options:

(A) the acceleration of $S$ is always directed towards the centre of the earth

(B) the angular momentum of $S$ about the centre of the earth changes in direction, but its magnitude remains constant

(C) the total mechanical energy of $S$ varies periodically with time

(D) the linear momentum of $S$ remains constant in magnitude

Sigma Percentile · Accepted Answer

### Introduction to Central Forces

When we look up at the night sky, the motion of celestial bodies seems like a perfectly choreographed cosmic dance.

But behind this beauty lies the rigorous and elegant framework of classical mechanics.

In this problem, we analyze a satellite $S$ of mass $m$ orbiting the Earth of mass $M$ in an elliptical path.

We are given that the mass of the satellite is extremely small compared to the Earth ($m \ll M$), which allows us to safely assume that the Earth remains stationary at one of the foci of the ellipse.

Let us dissect each option systematically to understand the underlying physics.

---

### Analyzing the Gravitational Force and Acceleration

According to Newton's Law of Universal Gravitation, the force exerted by the Earth on the satellite is given by:

$$\vec{F}_g = -\frac{GMm}{r^2} \hat{r}$$

Here, $\vec{r}$ is the position vector of the satellite pointing from the center of the Earth to the satellite, and $\hat{r}$ is the unit vector in that direction.

The negative sign indicates that the force is attractive, pulling the satellite directly toward the center of the Earth.

Using Newton's second law of motion, we can find the acceleration $\vec{a}$ of the satellite:

$$\vec{a} = \frac{\vec{F}_g}{m} = -\frac{GM}{r^2} \hat{r}$$

Since the gravitational force is always directed toward the center of the Earth, the acceleration vector $\vec{a}$ must also **always point toward the center of the Earth**.

This confirms that **Option (a) is correct**.

---

### The Conservation of Angular Momentum

Let's evaluate the torque $\vec{	au}$ acting on the satellite about the center of the Earth:

$$\vec{	au} = \vec{r} 	imes \vec{F}_g$$

Since the force vector $\vec{F}_g$ is directed along the line joining the satellite and the Earth's center, it is collinear with the position vector $\vec{r}$.

The angle between $\vec{r}$ and $\vec{F}_g$ is $180^\circ$, making their cross product zero:

$$\vec{	au} = \vec{r} 	imes \left(-\frac{GMm}{r^2} \hat{r}ight) = 0$$

Since the net torque acting on the satellite is zero, its angular momentum $\vec{L}$ must be conserved in both magnitude and direction:

$$\frac{d\vec{L}}{dt} = \vec{	au} = 0 \implies \vec{L} = 	ext{constant}$$

This means that the angular momentum vector is completely constant throughout the orbit.

Option (b) states that the direction of angular momentum changes, which is **incorrect**.

---

### Conservation of Mechanical Energy

The gravitational force is a conservative force.

In any conservative force field, the total mechanical energy $E$ of the system (which is the sum of its kinetic energy $K$ and gravitational potential energy $U$) remains constant:

$$E = K + U = \frac{1}{2}mv^2 - \frac{GMm}{r} = 	ext{constant}$$

While kinetic energy and potential energy vary individually as the distance $r$ changes, their sum remains strictly constant.

Therefore, the total mechanical energy does not vary periodically with time, making **Option (c) incorrect**.

---

### Variation of Linear Momentum

In an elliptical orbit, the distance $r$ of the satellite from the Earth is constantly changing.

By Kepler's second law (or the conservation of angular momentum), the speed of the satellite must vary:

* It is highest at the **perigee** (the point of closest approach).
* It is lowest at the **apogee** (the point of farthest distance).

Since the speed $v$ of the satellite changes continuously, the magnitude of its linear momentum $p = mv$ must also change periodically.

Thus, the linear momentum does not remain constant in magnitude, making **Option (d) incorrect**.

### Summary

Through systematic analysis of central forces, torque, energy, and momentum, we conclude that only the acceleration vector maintains a constant directional characteristic—always pointing toward the center of the Earth.

Hence, the correct option is **(a)**.

Animated Solution for Physics - Gravitation: A satellite $S$ is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth. Then,

Select Answer:

Visualized Solution

Visualizing the Elliptical Orbit

Analyzing the Gravitational Force

Determining the Direction of Acceleration

Evaluating Torque about the Earth's Center

Checking Angular Momentum (Option b)

Checking Mechanical Energy (Option c)

Checking Linear Momentum (Option d)

Conclusion

The Sigma Insight: Orbital Motion of a Satellite

Introduction to Central Forces

Analyzing the Gravitational Force and Acceleration

The Conservation of Angular Momentum

Conservation of Mechanical Energy

Variation of Linear Momentum

Summary

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