The escape velocity for a body projected from a planet depends on

Options

A. mass of the body
B. angle of projection
C. mass of the planet
D. radius of the body

Answer

To determine what the escape velocity for a body projected from a planet depends on, we can derive the formula for escape velocity and analyze its components. Here’s a step-by-step solution:

### Step 1: Understand Escape Velocity
Escape velocity is the minimum velocity required for an object to break free from the gravitational attraction of a planet without any further propulsion.

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

### Step 3: Kinetic Energy
The kinetic energy (K.E) of the object when it is projected with an initial velocity \( v_e \) (escape velocity) is given by:
\[ K.E = \frac{1}{2}mv_e^2 \]

### Step 4: Setting Up the Energy Conservation Equation
For the object to escape the gravitational field, the kinetic energy must be equal to the magnitude of the gravitational potential energy:
\[ \frac{1}{2}mv_e^2 = \frac{GMm}{r} \]

### Step 5: Canceling Mass
Since the mass \( m \) of the object appears on both sides of the equation, we can cancel it out:
\[ \frac{1}{2}v_e^2 = \frac{GM}{r} \]

### Step 6: Solving for Escape Velocity
Now, we can solve for \( v_e \):
\[ v_e^2 = \frac{2GM}{r} \]
Taking the square root of both sides gives us the formula for escape velocity:
\[ v_e = \sqrt{\frac{2GM}{r}} \]

### Step 7: Analyzing the Dependencies
From the derived formula, we can see that:
- Escape velocity \( v_e \) depends on the mass of the planet \( M \).
- Escape velocity \( v_e \) depends on the radius of the planet \( r \).
- Escape velocity does **not** depend on the mass of the object \( m \) or the angle of projection.

### Conclusion
The escape velocity for a body projected from a planet depends on the mass of the planet and the radius of the planet.