The escape velocity of a body depeds upon mass as

Options

A. `m^0`
B. `m`
C. `m^2`
D. `m^3`

Explanation

To determine how the escape velocity of a body depends on its mass, we can derive the formula for escape velocity using the principles of energy conservation. Here’s a step-by-step solution:

### Step 1: Understand the Concept of Escape Velocity
Escape velocity is the minimum velocity an object must have to break free from the gravitational pull of a celestial body without any further propulsion.

### Step 2: Write the Formula for Gravitational Potential Energy
The gravitational potential energy (U) of a mass \( m \) at a distance \( r \) from the center of a mass \( M \) (like the Earth) is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant.

### Step 3: Set Up the Conservation of Energy Equation
For an object to escape the gravitational field, its total mechanical energy must be zero or greater. Initially, the object has potential energy and no kinetic energy. At the point of escape, we want the potential energy to be zero (i.e., the object is at infinity). Thus, we set up the equation:
\[ \text{Initial Kinetic Energy} + \text{Initial Potential Energy} = \text{Final Kinetic Energy} + \text{Final Potential Energy} \]
This simplifies to:
\[ \frac{1}{2}mv^2 - \frac{GMm}{r} = 0 \]

### Step 4: Solve for Escape Velocity
Rearranging the equation gives:
\[ \frac{1}{2}mv^2 = \frac{GMm}{r} \]
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ \frac{1}{2}v^2 = \frac{GM}{r} \]
Multiplying both sides by 2:
\[ v^2 = \frac{2GM}{r} \]
Taking the square root:
\[ v = \sqrt{\frac{2GM}{r}} \]

### Step 5: Analyze the Dependence on Mass
From the derived formula for escape velocity \( v = \sqrt{\frac{2GM}{r}} \), we can see that:
- The escape velocity \( v \) depends on the mass \( M \) of the celestial body (like the Earth) and the radius \( r \) from the center of that body.
- Importantly, the mass \( m \) of the escaping object does not appear in the final expression for escape velocity, indicating that escape velocity is independent of the mass of the object trying to escape.

### Conclusion
The escape velocity of a body is independent of its mass. Therefore, the escape velocity does not depend on the mass of the object.

### Final Answer
The escape velocity of a body depends upon mass as \( m^0 \) (independent of mass).

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