Problem 769 - JEE Physics

Question

A satellite revolves around the earth in an elliptical orbit. Its speed (A) is the same at all points in the orbit (B) is greatest when it is closest to the earth (C) is greatest when it is farthest to the earth (D) goes on increasing or decreasing continuously depending upon the mass of the satellite

Step-by-Step Solution

Answer

The correct answer is (B) The satellite's speed is greatest when it is closest to Earth and smallest when it is farthest from Earth. This conclusion is based on Kepler's Second Law of Planetary Motion and the conservation of angular momentum, which states that the product of the satellite's mass, speed, and distance from Earth remains constant throughout its orbit.

Step 1: Review Kepler's Second Law of Planetary Motion

According to Kepler's Second Law, a satellite (or planet) revolving around a central body will sweep out equal areas in equal intervals of time. This means the satellite will move faster when it is closer to Earth and slower when it is farther away. This principle is related to the conservation of angular momentum. #Step 2: Describe the conservation of angular momentum#

Step 2: Conservation of Angular Momentum

The conservation of angular momentum states that the total angular momentum of an object remains constant if no external torques act on it. For a satellite in an elliptical orbit, its angular momentum, L, can be defined as the product of its mass (m), its speed (v), and its distance from the central body (r). Mathematically, this can be expressed as: \[ L = mvr \] Since the angular momentum is conserved, the product of the satellite's mass, speed, and distance from Earth remains constant throughout its orbit. #Step 3: Relate satellite speed to its distance from Earth#

Step 3: Using Conservation of Angular Momentum to Determine Speed

Because the angular momentum is constant, the product of the satellite's speed and distance from Earth, vr, must remain constant. That is, \[ vr = constant \] To determine the satellite's speed when it is closest to and farthest from Earth, we can use the relation: \[ v_{closest} r_{closest} = v_{farthest} r_{farthest} \] Since the distance from Earth is smaller when the satellite is closest to Earth, the satellite's speed must be greater to keep the product constant. Conversely, when the satellite is farther from Earth, its speed must be smaller. #Step 4: Choose the correct answer#

Step 4: Choose the Correct Answer

Based on our analysis, the correct answer is: (B) The satellite's speed is greatest when it is closest to Earth and smallest when it is farthest from Earth This conclusion is consistent with Kepler's Second Law and the conservation of angular momentum.

Key Concepts

Conservation of Angular Momentum

The conservation of angular momentum is a fundamental principle in physics and is particularly significant in understanding satellite and planetary motion. Angular momentum refers to the quantity of rotation an object has, taking into account its mass, speed, and distance from the point it rotates around. For an object in motion around a central body, like a satellite orbiting Earth, angular momentum is given by the formula:
\[ L = mvr \]where:

• \( m \) is the mass of the satellite,
• \( v \) is its orbital speed,
• \( r \) is the distance from the Earth.

The key aspect of the conservation of angular momentum is that if no external forces or torques are applied, this quantity remains constant over time. As a result, when a satellite moves in its orbital path, any changes in its speed or distance must adjust to ensure the product \( mvr \) stays the same.
The practical implication is seen in the varying speeds of satellites. As a satellite moves closer to Earth in its elliptical path, its distance \( r \) decreases, requiring its speed \( v \) to increase to keep \( L \) constant. Conversely, as it moves further from Earth, \( v \) must decrease as \( r \) increases.

Elliptical Orbits

Elliptical orbits are one of the pathways that celestial bodies follow when revolving around a central object. Unlike circular orbits where the distance from the central body remains constant, elliptical orbits vary in distance, creating a unique set of motions observed in planetary and satellite movement.
An ellipse is a shape that resembles a flattened circle, and it is characterized by two points called foci, one of which is occupied by the central body, such as Earth. Kepler's First Law of Planetary Motion states that the orbits of planets (and satellites) are ellipses with the central mass located at one of these foci.
This elliptical path means that a satellite will sometimes be much closer to the planet (the point called the perigee) and at other times much further away (the point known as the apogee). This varying distance affects the satellite's speed due to the conservation of angular momentum. At perigee, the satellite speeds up, and at apogee, it slows down.
Ellipses are defined by their semi-major and semi-minor axes, representing the longest and shortest diameters, respectively. The dynamic nature of elliptical orbits is fundamental in predicting orbital mechanics and understanding how celestial bodies interact in space.

Planetary Motion

Planetary motion describes the movements of celestial bodies around a star or planet, often following paths defined by Kepler's Laws. These laws collectively provide insights into how planets and satellites behave in their orbits.
Kepler's First Law states that planets move in elliptical orbits, while the Second Law, also known as the Law of Equal Areas, asserts that a line connecting a planet to the sun sweeps out equal areas in equal times. This results in varying speeds of celestial objects in their orbits.

• Closer to the central body, planets move faster.
• Farther away, they move slower.

Kepler's Third Law relates the time a planet takes to complete one orbit with its average distance from the central body, indicating that the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.
These laws are not only applicable to planets in our solar system but also to moons, comets, and artificial satellites. By observing these motions, astronomers can predict positions, design space missions, and understand gravitational interactions between multiple celestial bodies. Understanding planetary motion provides a foundation for modern astrophysics and space exploration.