Problem 742 - JEE Physics

Question

There are two planets, the ratio of radius of two planets is \(\mathrm{k}\) but the acceleration due to gravity of both planets are \(\mathrm{g}\) what will be the ratio of their escape velocity. (A) \((\mathrm{kg})^{1 / 2}\) (B) \((\mathrm{kg})^{-1 / 2}\) (C) \((\mathrm{kg})^{2}\) (D) \((\mathrm{kg})^{-2}\)

Step-by-Step Solution

Answer

The short answer is: (A) \((kg)^{1/2}\).

Step 1: Write down the escape velocity formula

The escape velocity formula 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 radius of the planet.

Step 2: Find the ratio of masses of the planets

We are given that the ratio of the radii of the planets is \(k\). We know that the acceleration due to gravity on both planets is the same, so we can write: \[g = \frac{GM_1}{R_1^2} = \frac{GM_2}{R_2^2}\] Dividing the equations and using the given ratio of radii, we get: \[\frac{M_1}{M_2} = \frac{R_1^2}{R_2^2} = k^2\]

Step 3: Write down the escape velocities for both planets

Using the escape velocity formula and the ratio of masses found in the previous step, we can write the escape velocities for both planets as: \[v_{e1} = \sqrt{\frac{2GM_1}{R_1}}\] \[v_{e2} = \sqrt{\frac{2GM_2}{R_2}}\]

Step 4: Find the ratio of escape velocities

Now, we will find the ratio of escape velocities by dividing the equations: \[\frac{v_{e1}}{v_{e2}} = \frac{\sqrt{\frac{2GM_1}{R_1}}}{\sqrt{\frac{2GM_2}{R_2}}}\] Using the ratios found in Step 2, we can simplify this expression: \[\frac{v_{e1}}{v_{e2}} = \sqrt{\frac{M_1}{M_2} \cdot \frac{R_2}{R_1}} = \sqrt{\frac{k^2}{k}} = \sqrt{k}\]

Step 5: Identify the correct answer

Comparing our final answer to the provided options, we find that the correct answer is: (A) \((kg)^{1/2}\).

Key Concepts

Planetary Motion

Planetary motion refers to the orbital movements of planets around stars, primarily due to the gravitational pull. Understanding this concept is crucial for explaining how planets, including Earth, are held in their orbits. This motion is governed by Newton's laws of motion and universal gravitation.
From a physics standpoint:

• Newton's First Law: An object in motion will stay in motion unless acted upon by an external force. Planets move in their orbits because they maintain a balance between gravitational pull and their inertia.
• Newton's Second Law: Acceleration is produced when a force acts on a mass. Gravitational force contributes to a planet's orbital path.
• Kepler’s Laws: These laws describe the orbits of planets as ellipses, with varying speed depending on their position.

Overall, planetary motion explains the regular paths planets take, the effects of gravity, and how they maintain stable orbits. Understanding this helps determine concepts like escape velocity, which is essential to comprehend if an object can leave a planet's gravitational influence.

Acceleration Due to Gravity

Acceleration due to gravity, denoted as "g", is the rate at which an object accelerates when falling freely under the influence of a planet's gravitational force. It plays a vital role in understanding planetary environments and how they affect object movements.

• Constant Value on Earth: On Earth's surface, "g" is approximately 9.81 m/s², but this value can vary slightly in different locations.
• Dependence on Mass and Radius: The formula for gravitational acceleration is given by \( g = \frac{GM}{R^2} \). Here, \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius. This implies that "g" is directly related to a planet's mass and inversely to the square of its radius.
• Implications for Other Planets: On different planets, "g" changes because their masses and radii differ. For two planets with the same "g", their mass-to-radius relationships must be adjusted accordingly.

Knowing the acceleration due to gravity is essential for calculating other crucial physics concepts, like escape velocity, which depends on a planet's gravity to determine the speed needed for an object to break free from it.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a crucial component in the universal law of gravitation. It characterizes the strength of gravity for any pair of masses. Understanding \( G \) is fundamental in astronomy, physics, and engineering.

• Value and Units: The gravitational constant \( G \) has a value of approximately \( 6.674 \times 10^{-11} \text{N} \times \text{m}^2/\text{kg}^2 \).
• Role in Gravitational Force: It appears in the equation \( F = \frac{GMm}{R^2} \), where \( F \) is the gravitational force between two masses \( M \) and \( m \), and \( R \) is the distance between them. \( G \) is key in calculating how strong the gravitational pull is.
• Impact on Planetary Calculations: \( G \) is a fundamental constant in determining earth-based and astronomical calculations, like the forces exerted by satellites and planets in motion.

In essence, the gravitational constant serves as the universal "glue" in the cosmos, providing a measure of how gravity operates across different contexts. It's indispensable for accurately understanding force interactions in planetary systems.