Problem 80 - Master Resource Book in JEE Main Physics

Question

If \(v_{e}\) and \(v_{o}\) represent the escape velocity and orbital velocity of satellite corresponding to a circular orbit of radius \(R\), then (a) \(v_{e}=v_{o}\) (b) \(\sqrt{2} v_{o}=v_{e}\) (c) \(v_{e}=\frac{v_{o}}{\sqrt{2}}\) (d) \(v_{e}\) and \(v_{o}\) are not related

Step-by-Step Solution

Answer

(b) \(\sqrt{2} v_o = v_e\)

Step 1: Understanding Escape and Orbital Velocity

Escape velocity (v_e) is the minimum velocity required for an object to escape a celestial body's gravitational field without further propulsion. Orbital velocity (v_o) is the speed needed to keep an object in circular orbit around the body. Both depend on gravitational potential energy and kinetic energy.

Step 2: Formula for Escape Velocity

The formula for escape velocity is given by: \( v_e = \sqrt{\frac{2GM}{R}} \), where M is the mass of the celestial body, G is the gravitational constant, and R is the radius of the orbit.

Step 3: Formula for Orbital Velocity

The formula for orbital velocity is: \( v_o = \sqrt{\frac{GM}{R}} \) using the same variables as the escape velocity formula.

Step 4: Relating Escape Velocity to Orbital Velocity

By comparing the formulas, we see that \( v_e = \sqrt{2} v_o \). This means that escape velocity is \(\sqrt{2}\) times the orbital velocity.

Key Concepts

Orbital Velocity

Orbital velocity is the speed an object needs to maintain a stable path around a celestial body. Imagine a satellite circling the Earth without falling back or flying away. For this, the balance between gravitational pull and the object’s speed is crucial.
This velocity depends on two key factors: the mass of the celestial body and the radius of the circular orbit. The formula to calculate orbital velocity is \[ v_o = \sqrt{\frac{GM}{R}} \]where:

• \( G \) is the gravitational constant,
• \( M \) is the mass of the celestial body,
• \( R \) is the radius of the orbit.

This ensures that centripetal force keeps the object moving in a circle.
When satellites achieve this velocity, they remain in orbit without requiring additional energy inputs because the gravitational force and centripetal acceleration precisely balance each other.

Gravitational Potential Energy

Gravitational potential energy refers to the energy stored in an object due to its position in a gravitational field. It is essential when discussing objects in orbit or those attempting to leave a planet's gravitational pull.
The formula for gravitational potential energy is:\[ U = -\frac{GMm}{R} \] where:

• \( G \) is the gravitational constant,
• \( M \) is the mass of the celestial body,
• \( m \) is the mass of the object, such as a satellite,
• \( R \) is the distance from the center of the celestial body.

The negative sign indicates that energy is required to move the object to an infinite distance away (escape the gravitational field).
This energy is crucial when considering both orbital and escape velocities since altering an object's position in the gravitational field requires considerable energy.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It plays a vital role in understanding the forces acting on objects in orbit or attempting to escape a gravitational field. The formula for kinetic energy is:\[ KE = \frac{1}{2}mv^2 \] where:

• \( m \) is the mass of the object,
• \( v \) is its velocity.

Objects in orbit, such as satellites, require sufficient kinetic energy to counteract gravitational forces and remain aloft. Therefore, their kinetic energy is linked directly to their orbital velocity.
When it comes to escaping velocity, the object’s kinetic energy should be high enough to not only stay in orbit but also overcome gravitational forces completely. Hence, understanding kinetic energy is critical for calculating how fast and with how much energy an object must move to achieve its required path or escape a gravitational pull.

Circular Orbit

A circular orbit occurs when an object moves around another in a perfect circle. This kind of orbit means that the distance from the object to the center of the celestial body remains constant.
For an object to maintain a circular orbit, two main forces must be in balance:

• Gravitational force pulling the object towards the celestial body,
• Centripetal force, the inward force needed to keep the object moving in a circle.

The object's velocity in a circular orbit must match its orbital velocity to ensure it neither falls into the celestial body nor escapes into space.
This is significant for satellites and space missions that rely on maintaining stable orbits for functionality and communication. Understanding circular orbits helps in planning trajectory paths and ensuring that human-made objects in space remain where they are needed.