What is the gravitational potential energy (MJ) of a body of mass 3 kg on a planet where escape velocity is `4 km s^(-1)` ?

Explanation

To find the gravitational potential energy of a body of mass 3 kg on a planet where the escape velocity is 4 km/s, we can follow these steps:

### Step-by-Step Solution

1. **Convert Escape Velocity to SI Units**:
The escape velocity is given as 4 km/s. We need to convert this to meters per second (m/s):
\[ V = 4 \, \text{km/s} = 4 \times 1000 \, \text{m/s} = 4000 \, \text{m/s} \]

2. **Understand the Relationship Between Escape Velocity and Gravitational Potential Energy**:
The escape velocity \( V \) is related to the gravitational potential energy \( U \) by the formula:
\[ V = \sqrt{\frac{2GM}{R}} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

3. **Express Gravitational Potential Energy**:
The gravitational potential energy \( U \) of a mass \( m \) in the gravitational field of a planet is given by:
\[ U = -\frac{GMm}{R} \]
We can derive a relationship between \( U \) and \( V \):
\[ U = -\frac{1}{2} m V^2 \]

4. **Substitute Values into the Potential Energy Formula**:
Now, substituting the mass \( m = 3 \, \text{kg} \) and the escape velocity \( V = 4000 \, \text{m/s} \):
\[ U = -\frac{1}{2} \times 3 \, \text{kg} \times (4000 \, \text{m/s})^2 \]

5. **Calculate the Potential Energy**:
First, calculate \( (4000)^2 \):
\[ (4000)^2 = 16000000 \, \text{m}^2/\text{s}^2 \]
Now substitute this back into the equation for \( U \):
\[ U = -\frac{1}{2} \times 3 \times 16000000 \]
\[ U = -\frac{3 \times 16000000}{2} = -24000000 \, \text{J} \]

6. **Convert Joules to Megajoules**:
To convert joules to megajoules, we divide by \( 10^6 \):
\[ U = -24000000 \, \text{J} = -24 \, \text{MJ} \]

### Final Answer:
The gravitational potential energy of the body is \( -24 \, \text{MJ} \).

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