Escape velocity of a body of 1 g mass on a planet is 100 m/sec . Gravitational Potential energy of the body at the planet is

Options

A. ` - 5000J`
B. `- 1000 J`
C. `- 2400J`
D. `5000 J`

Explanation

To find the gravitational potential energy (U) of a body of mass 1 kg on a planet where the escape velocity is 100 m/s, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the body (m) = 1 kg
- Escape velocity (v_e) = 100 m/s

2. **Use the formula for escape velocity:**
The escape velocity (v_e) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where:
- G = universal gravitational constant (approximately \(6.674 \times 10^{-11} \, \text{m}^3/\text{kg s}^2\))
- M = mass of the planet
- R = radius of the planet

3. **Square both sides of the escape velocity equation:**
\[ v_e^2 = \frac{2GM}{R} \]
Substituting the value of escape velocity:
\[ (100 \, \text{m/s})^2 = \frac{2GM}{R} \]
\[ 10000 = \frac{2GM}{R} \]

4. **Rearranging the equation to find \( \frac{GM}{R} \):**
\[ \frac{GM}{R} = \frac{10000}{2} = 5000 \]

5. **Use the formula for gravitational potential energy:**
The gravitational potential energy (U) of the body at the surface of the planet is given by:
\[ U = -\frac{GMm}{R} \]

6. **Substituting the value of \( \frac{GM}{R} \) into the potential energy formula:**
\[ U = -\left(5000 \, \text{m}^3/\text{kg s}^2\right) \times (1 \, \text{kg}) \]
\[ U = -5000 \, \text{Joules} \]

### Final Answer:
The gravitational potential energy of the body at the planet is \(-5000 \, \text{Joules}\).

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