Problem 64 - A Complete Resource Book in Physics for JEE Main

Question

The escape velocity of a body depends upon mass as (A) \(m^{0}\) (B) \(m^{1}\) (C) \(m^{2}\) (D) \(m^{3}\)

Step-by-Step Solution

Answer

The escape velocity depends on the mass of a body as \(M^1\). Therefore, the correct answer is option (B): \(m^1\).

Step 1: Write down the formula for escape velocity.

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 celestial body, and \(r\) is the distance from its center. In this case, we are concerned with how the escape velocity depends on the mass of the body. We can observe that in the formula, mass is in the numerator and raised to the power of 1.

Step 2: Analyze the relation between escape velocity and mass.

From the escape velocity formula, we can see that the mass of the celestial body, \(M\), has a power of 1 as it is directly multiplied in the formula: \[ v_e = \sqrt{\frac{2G*(M^1)}{r}} \] Thus, the escape velocity depends on mass as \(M^1\).

Step 3: Match the relation with the given options.

We found that the escape velocity depends on the mass of the body as \(M^1\). Now, we will match this with the given options: (A) \(m^0\) (B) \(m^1\) (C) \(m^2\) (D) \(m^3\) As we can see, the correct answer is option (B): \(m^1\).