Problem 741 - JEE Physics

Question

The escape velocity of a planet having mass 6 times and radius 2 times as that of earth is (A) \(\sqrt{3} \mathrm{~V}_{\mathrm{e}}\) (B) \(3 \mathrm{~V}_{\mathrm{e}}\) (C) \(\sqrt{2} \mathrm{~V}_{\mathrm{e}}\) (D) \(2 \mathrm{~V}_{\mathrm{e}}\)

Step-by-Step Solution

Answer

The escape velocity of a planet with a mass 6 times and a radius 2 times greater than Earth's is (A) \(\sqrt{3} V_{e}\).

Step 1: Understand the Escape Velocity Formula

Escape velocity is the minimum speed an object needs to break free from a celestial body's gravitational pull. The escape velocity formula is given by: \[v_{esc} = \sqrt{\frac{2GM}{r}}\] where \(v_{esc}\) is the escape velocity, \(G\) is the gravitational constant, \(M\) is the mass of the celestial body, and \(r\) is its radius.

Step 2: Find the Escape Velocity of Earth

We don't have the values of the gravitational constant, mass of the Earth, and Earth's radius. However, we don't need them since we only need to find the ratio between the other planet's escape velocity and Earth's escape velocity. Let's denote Earth's escape velocity as \(V_{e}\).

Step 3: Apply the Values of the Given Planet

The given planet has a mass of \(6M\) and a radius of \(2r\) (where M and r are Earth's mass and radius). We can plug these values into the formula for escape velocity: \[v_{esc}^{planet} = \sqrt{\frac{2G(6M)}{2r}}\]

Step 4: Simplify the Equation

Now, we can simplify the equation: \[v_{esc}^{planet} = \sqrt{\frac{12GM}{2r}} = \sqrt{\frac{6GM}{r}}\]

Step 5: Calculate the Ratio of the Given Planet's Escape Velocity to Earth's Escape Velocity

Finally, we need to find the ratio between the planet's escape velocity and Earth's escape velocity: \[\frac{v_{esc}^{planet}}{V_{e}} = \frac{\sqrt{\frac{6GM}{r}}}{\sqrt{\frac{2GM}{r}}}\] The mass \(M\) and radius \(r\) are common in both the numerator and the denominator, thus will cancel out. We are left with: \[\frac{v_{esc}^{planet}}{V_{e}} = \sqrt{\frac{6}{2}} = \sqrt{3}\] The answer is (A) \(\sqrt{3} V_{e}\).

Key Concepts

Gravitational Force

Gravitational force is a fundamental interaction in nature that pulls objects towards each other. It's best described by Isaac Newton's law of universal gravitation, which states that every mass attracts every other mass through the force of gravity. This force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula illustrating this is:\[F = \frac{Gm_1m_2}{r^2}\]

• \(F\) is the gravitational force.
• \(G\) is the gravitational constant \((6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2)\).
• \(m_1\) and \(m_2\) are the masses of the two objects.
• \(r\) is the distance between the centers of the two masses.

In the context of escape velocity, gravitational force is crucial because it determines how much energy is needed for an object to overcome this pull. It explains why planets with more mass or smaller radii generally have higher escape velocities: more mass increases the force, and a smaller radius decreases the distance at which the force is applied.

Celestial Mechanics

Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of celestial bodies like planets, moons, and stars. One of its primary purposes is to predict the future positions of these objects by understanding the forces acting upon them. When considering escape velocity from a celestial object, celestial mechanics assists in understanding how objects in space influence each other. The escape velocity depends on:

• The gravitational pull of the object which an entity is trying to escape from.
• The object's radius, which defines how close one can be to the center of its gravity while on the surface.

These aspects all contribute to the calculations involving the kinetic and potential energy of a body. The built-up kinetic energy must be sufficient to exceed the gravitational potential energy for the object to "escape." Consider an object moving in space; to stay in motion, it must overcome the gravitational forces analyzed through celestial mechanics.

Planetary Science

Planetary science is the study of planets, their moons, and planetary systems, in particular those of the solar system and the processes that form them. Scientists study planetary compositions, atmospheres, and phenomena. Understanding escape velocity is important in planetary science because it provides insight into a planet's atmosphere and potential to hold it. Escape velocity must be higher than the average speed of gas particles in a planet's atmosphere; otherwise, those particles will escape into space over time.

• Planet Mass: More mass typically means a stronger gravitational pull, indicating a higher escape velocity.
• Planet Radius: A smaller radius means that the gravitational pull extends closer to the surface, also affecting escape velocity.

Analyzing these factors helps scientists understand atmospheric retention and the potential for planets to support life. They reveal why certain celestial bodies, like Earth, can maintain an atmosphere, while others, such as the Moon, cannot. This scientific knowledge is central to understanding the past, present, and future potential of different planets in our solar system and beyond.