Problem 724 - JEE Physics

Question

Escape velocity of a body of \(1 \mathrm{~kg}\) on a planet is \(100 \mathrm{~ms}^{-1}\). Gravitational potential energy of the body at the planet is \(=\) \(\begin{array}{ll}\text { (A) } \overline{-5000} & \text { (B) }-1000\end{array}\) (C) \(-2400\) (D) 5000

Step-by-Step Solution

Answer

The gravitational potential energy of the body at the planet's surface is \(-5000\mathrm{~J}\). So, the correct answer is (A) \(-5000\).

Step 1: Formula for escape velocity

To find the gravitational potential energy, we will first look at the formula for escape velocity. The escape velocity is given by: \(v_e = \sqrt{\frac{2GM}{r}}\) where \(v_e\) is the escape velocity, \(G\) is the gravitational constant, \(M\) is the mass of the planet, and \(r\) is the distance from the center of the planet to the body. Using the given escape velocity, \(100\mathrm{~ms^{-1}}\), we can calculate the value of \(\frac{2GM}{r}\).

Step 2: Kinetic energy and gravitational potential energy

Next, we can relate kinetic energy (KE) and gravitational potential energy (GPE) to escape velocity. For a body to escape a planet's gravitational field, its kinetic energy needs to be equal to the gravitational potential energy: \(KE = GPE\) The kinetic energy of the body is given by: \(KE = \frac{1}{2}mv^2\) Using the escape velocity, the kinetic energy of the body is: \(KE = \frac{1}{2}(1\textrm{ kg})(100\textrm{ ms}^{-1})^2\) Now, the gravitational potential energy is: \(GPE = -\frac{GMm}{r}\) Since \(KE = GPE\), we have: \(\frac{1}{2}mv^2 = -\frac{GMm}{r} \Rightarrow \frac{1}{2}v^2 = -\frac{GM}{r}\)

Step 3: Calculate the gravitational potential energy

Now we can solve for \(GPE\): \(-\frac{GM}{r} = \frac{1}{2}(100\textrm{ ms}^{-1})^2\) Let's plug in the value of \(\frac{2GM}{r}\) we got in step 1: \(GPE = -\frac{1}{2}(100\textrm{ ms}^{-1})^2 = -5000\mathrm{~J}\) The gravitational potential energy of the body at the planet's surface is -5000 J. So, the correct answer is (A) \(-5000\).

Key Concepts

Escape Velocity

Escape velocity is a crucial concept in planetary physics. It refers to the minimum speed required for an object, such as a spacecraft, to break free from a planet's gravitational pull without any further propulsion.
This speed ensures that the object can overcome the gravitational forces and escape into space without falling back to the planet's surface.The formula for escape velocity is given by:

• \(v_e = \sqrt{\frac{2GM}{r}}\)

Here, \(v_e\) represents the escape velocity, \(G\) is the universal gravitational constant, \(M\) is the mass of the planet, and \(r\) is the radius or distance from the planet's center to the object.
Notice that escape velocity depends on both the mass of the planet and the distance from its center.
Larger planets with more mass or greater distances will have higher escape velocities.

Kinetic Energy

Kinetic energy (KE) is the energy that an object possesses due to its motion. It is directly linked to the mass and velocity of the object.
The formula for kinetic energy is:

• \(KE = \frac{1}{2}mv^2\)

Where \(m\) stands for the mass of the object and \(v\) is its velocity.
In this context, kinetic energy is essential because it helps us understand the energy needed to reach escape velocity.
For an object to leave a planet, its kinetic energy must be equal to the gravitational potential energy (GPE).
This equality shows how gravitational forces balance kinetic energy to determine if an object can escape the planet's gravitational field.

Gravitational Constant

The gravitational constant, symbolized by \(G\), is a fundamental constant that appears in Newton's law of universal gravitation.
It quantifies the strength of the gravitational force between two masses separated by a distance.The value of the gravitational constant is approximately:

• \(G = 6.674 \times 10^{-11} \text{ m}^3 \text{kg}^{-1} \text{s}^{-2}\)

This constant is crucial in the formula for escape velocity since it helps determine the gravitational pull a planet exerts on an object.
Without \(G\), we would not be able to calculate the force between celestial bodies, making it a vital part of planetary physics. The gravitational constant ensures that calculations involving gravity are consistent and accurate.

Planetary Physics

Planetary physics is a branch of physics that deals with understanding the forces and behaviors of planets in our universe.
It involves studying planet formations, gravitational interactions, and the movement of celestial bodies.
Escape velocity, kinetic energy, and gravitational potential are significant aspects within this field, as they explain the mechanics of how objects interact with planets and their gravitational fields. The study of planetary physics helps scientists predict and understand the motions of planets and other astronomical objects such as moons and asteroids.
By combining theories and mathematical equations, planetary physics allows us to explore and visualize dynamics beyond our planet, making it crucial for space exploration and our understanding of the universe.