Problem 731 - JEE Physics

Question

The escape velocity for a sphere of mass \(\mathrm{m}\) from earth having mass \(\mathrm{M}\) and Radius \(\mathrm{R}\) mass is given by (A) \(\sqrt{[}(2 \mathrm{GM}) / \mathrm{R}]\) (B) \(2 \sqrt{(\mathrm{GM} / \mathrm{R})}\) (C) \(\sqrt{[}(2 \mathrm{GMm}) / \mathrm{R}]\) (D) \(\sqrt{(\mathrm{GM} / \mathrm{R})}\)

Step-by-Step Solution

Answer

The correct escape velocity formula for a sphere of mass \(m\) from Earth having mass \(M\) and radius \(R\) is (A) \(\sqrt{(2 \mathrm{GM}) / \mathrm{R}}\).

Step 1: Remember the definition of escape velocity

The escape velocity is the minimum speed an object must have to escape from the gravitational force of a planet or any celestial body. In this case, we are considering the object escaping from Earth's gravitational force.

Step 2: Apply conservation of energy principle

Conservation of energy principle states that the total mechanical energy (potential + kinetic) of the system remains constant. In this situation, the total mechanical energy (E) can be represented as: \[E = KE + PE\] Where KE is the kinetic energy and PE is the gravitational potential energy.

Step 3: Formulate Kinetic and Potential energy expressions

The Kinetic energy (KE) is given by: \[KE = \frac{1}{2}mv^2\] Where m is the mass of the object and v is its velocity. The gravitational potential energy (PE) is given by: \[PE = -\frac{GMm}{r}\] Where G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and r is the distance between the center of masses of both the object and Earth.

Step 4: Setting the total mechanical energy to zero and solving for escape velocity

Since the object is escaping the gravitational force, we can equate the total mechanical energy to zero. This will result in the following equation: \[E = KE + PE = 0\] Now, substitute the expressions for KE and PE from Step 3 into this equation: \[\frac{1}{2}mv^2 -\frac{GMm}{r} = 0\]

Step 5: Isolate the velocity in the equation

Now, we need to solve this equation for the escape velocity v. First, we can divide both sides of the equation by m: \[\frac{1}{2}v^2 -\frac{GM}{r} = 0\] Next, add \(\frac{GM}{r}\) to both sides: \[\frac{1}{2}v^2 = \frac{GM}{r}\] Now, multiply both sides by 2: \[v^2 = \frac{2GM}{r}\] Finally, take the square root: \[v = \sqrt{\frac{2GM}{r}}\] Comparing this escape velocity formula to the options given in the exercise, we can identify the correct answer: (A) \(\sqrt{[}(2 \mathrm{GM}) / \mathrm{R}]\)

Key Concepts

Conservation of Energy

The principle of "Conservation of Energy" is a fundamental concept in physics that states energy in a closed system can neither be created nor destroyed. It can only change forms, such as from kinetic energy to potential energy and vice versa. In the context of escape velocity, this principle is key. When an object moves from the surface of the Earth to escape its gravitational pull, its total energy remains constant throughout the process.

The total mechanical energy of a system combines both kinetic energy (energy of motion) and gravitational potential energy (energy stored due to an object's position in a gravitational field). As an object reaches escape velocity, these energies adjust accordingly to allow the object to break free from Earth's gravity. Ultimately, the sum of kinetic and potential energies equals zero at the point of escape, illustrating energy conservation in action.

• Kinetic Energy: Energy due to motion
• Gravitational Potential Energy: Energy due to position in a gravitational field

Understanding this principle helps explain why objects with enough initial speed can overcome the gravitational bind of Earth.

Kinetic Energy

"Kinetic Energy" is the energy an object possesses due to its motion. The formula for kinetic energy is expressed as \( KE = \frac{1}{2}mv^2 \), where \(m\) is mass and \(v\) is velocity. For escape velocity, kinetic energy must be sufficient to counteract gravitational potential energy. This ensures that the spacecraft, satellite, or object can leave Earth's gravitational field.

During launch, as the object's speed increases, so does its kinetic energy. When the object reaches the escape velocity threshold, its energy is enough to overcome gravitational constraints, allowing it to enter space. Therefore, kinetic energy is crucial for understanding how object motion powers their journey into space.

• Direct relation: More velocity increases kinetic energy
• Key role: Powers motion required to overcome gravity

By grasping kinetic energy's role, you understand how speed and energy help objects escape gravitational fields.

Gravitational Potential Energy

Gravitational Potential Energy (GPE) is the energy stored in an object due to its position in a gravitational field. It depends on the mass of the object, the mass of the Earth, and the distance between their centers. The formula for gravitational potential energy is \( PE = -\frac{GMm}{r} \), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, \(m\) is the mass of the object, and \(r\) is the distance from the center of the Earth.

In the escape velocity scenario, gravitational potential energy is what binds the object to Earth. It is negative because work would need to be done against gravity to move the object to that position. Therefore, when an object reaches escape velocity, its gravitational potential energy must be zero or less, effectively balancing kinetic energy to allow departure from Earth's gravitational influence.

• Negative value: Indicates work needed to move object
• Depends on mass and distance: Key in calculating escape potential

This concept can be better understood by visualizing how potential energy changes as you move farther away from Earth.

Gravitational Constant

The "Gravitational Constant," represented by \(G\), is a fundamental constant in physics that quantifies the strength of the gravitational force between two objects. The value of \(G\) is approximately \(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\). This value appears in the formula for the gravitational force as well as in the equation for gravitational potential energy.

In the context of escape velocity, \(G\) plays a crucial role when calculating the speed needed for an object to break away from the Earth's gravitational pull. The formula for escape velocity uses \(G\) to balance the forces at play between an object's kinetic energy and the potential energy holding it close to the Earth.

• Constant value: Ensures consistent gravitational calculations
• Integral in formulas: Helps determine escape velocity through potential balance

By understanding the gravitational constant, you can gain insight into how universally applicable principles guide object interactions in space.