Problem 38 - Master Resource Book in JEE Main Physics

Question

A body is imparted a velocity \(v\) from surface of the earth. If \(v_{0}\) is orbital velocity and \(v_{e}\) be the escape velocity, then for (a) \(v=v_{0}\), the body follows a circular track around the earth (b) \(v>v_{0}\), but \(v_{e}\), the body follows hyperbolic path and escapes the gravitational pull of the earth

Step-by-Step Solution

Answer

Correct options: (a), (b), (c), (d) based on given conditions.

Step 1: Understanding Orbital Velocity

The orbital velocity, denoted as \(v_0\), is the minimum velocity a body needs to move in a stable circular orbit around a massive body, like Earth, without additional propulsion. This velocity is calculated using \(v_0 = \sqrt{\frac{GM}{r}}\), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the radius from the center of the Earth to the body.

Step 2: Understanding Escape Velocity

The escape velocity, denoted as \(v_e\), is the minimum velocity required for a body to break free from the gravitational pull of a planet without any additional propulsion. It is given by \(v_e = \sqrt{2}\cdot \sqrt{\frac{GM}{r}}\), which is \(\sqrt{2}\) times the orbital velocity.

Step 3: Analyzing the First Case (\(v = v_0\))

When the body's velocity \(v\) equals the orbital velocity \(v_0\), the body will maintain a circular orbit around the Earth. Since the velocity is exactly the orbital speed, the gravitational force acts as the centripetal force needed for circular motion. Hence, option (a) "the body follows a circular track around the Earth" is correct.

Step 4: Analyzing the Second Case (\(v > v_0\) but \(v < v_e\))

If the body is imparted a velocity greater than the orbital velocity \(v_0\) but less than the escape velocity \(v_e\), the trajectory becomes elliptical. In this scenario, while it exceeds the speed needed for a stable orbit, it is insufficient to escape, leading to an elliptical trajectory that eventually brings the body back to Earth. Thus, option (b) "the body follows an elliptical path and returns to the surface of Earth" is correct.

Step 5: Analyzing the Third Case (\(v < v_0\))

When the velocity \(v\) is less than the orbital velocity \(v_0\), the gravitational pull will dominate, causing the trajectory to be elliptical, and it will descend back to Earth. Thus, option (c) "the body follows an elliptical path and returns to the surface of Earth" is accurate.

Step 6: Analyzing the Fourth Case (\(v > v_e\))

With a velocity greater than the escape velocity \(v_e\), the body's energy surpasses the gravitational binding energy, allowing it to escape Earth's gravity. The path becomes hyperbolic, and the body won't return to the Earth's surface. Therefore, option (d) "the body follows a hyperbolic path and escapes the gravitational pull of the Earth" is correct.

Key Concepts

Orbital Velocity

To maintain a stable orbit around Earth, a body must travel at a specific speed known as orbital velocity. This speed allows the gravitational pull of Earth to act as the centripetal force necessary for a circular motion. Without external propulsion, the object remains in orbit due to this delicate balance.
The orbital velocity, denoted as \( v_0 \), is determined by the formula:

• \( v_0 = \sqrt{\frac{GM}{r}} \)

Here, \( G \) is the gravitational constant, \( M \) is the mass of Earth, and \( r \) is the radius from Earth's center to the body. Understanding orbital velocity is crucial for planning satellite launches and predicting natural orbital paths.
For instance, if a satellite orbits too slowly, it will descend to Earth, while a faster speed could lead to an escape trajectory.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from Earth's gravitational field. Unlike orbital velocity, escape velocity ensures the object doesn't return; it can leave the gravitational influence entirely. This concept is vital for missions intending to reach other planets or moons.
The escape velocity \( v_e \) at Earth's surface is represented by:

• \( v_e = \sqrt{2} \cdot \sqrt{\frac{GM}{r}} \)

This means that escape velocity is \( \sqrt{2} \) times the orbital velocity.
To achieve this, additional propulsion is typically necessary. Just like throwing a ball high enough to "escape" your hand’s pull and fly out of sight, rockets must be powerful enough to push their payload past Earth's gravitational grasp. Understanding escape velocity is pivotal when planning interplanetary travel.

Gravitational Force

Gravitational force is the force of attraction between two masses. For an object orbiting Earth, this force pulls it towards the planet's center, providing the central force needed to maintain orbit.
The formula for gravitational force \( F \) between two bodies is:

• \( F = \frac{G M m}{r^2} \)

Here, \( M \) is Earth's mass, \( m \) is the object's mass, and \( r \) is the distance between their centers.
This universal force is what keeps the planets rotating around the Sun and the Moon around Earth. It is incredibly important in orbital mechanics and understanding how bodies in space move and interact.
The balance of gravitational force with velocity allows bodies to move in stable orbits without falling into one another.

Circular Orbit

A circular orbit occurs when a body travels around Earth at a constant speed equal to the orbital velocity. This keeps the object at a consistent distance from Earth, maintaining a circular path.
In a perfect circular orbit:

• Speed remains constant
• Distance from Earth doesn't change

The gravitational force acts as the centripetal force to keep the orbit intact, ensuring the body doesn't spiral away or fall back to Earth.
Circular orbits are especially useful for certain satellites like geostationary satellites, which need to maintain the same position relative to Earth's surface to provide consistent data.

Elliptical Trajectory

When a body has a velocity that is either more or less than the required orbital velocity but less than the escape velocity, it follows an elliptical trajectory. This path causes the object to oscillate between a point closest to Earth (perigee) and farthest (apogee).
In elliptical orbits:

• The velocity varies throughout the orbit
• Distance from Earth changes constantly

Elliptical trajectories are typical for many celestial bodies and most artificial satellites, which need to cover different parts of Earth's surface.
Understanding the factors that shape elliptical trajectories is essential for maneuvering satellites and planning missions that require capturing or avoiding certain celestial objects at different times.