Escape velocity of a body `1 kg` mass on a planet is `100 ms^(-1)`. Gravitational potential energy of the body at that planet is

Options

A. `- 5000 J`
B. `- 1000 J`
C. `-2400 J`
D. `-10000 J`

Answer

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

### Step 1: Understand the relationship between escape velocity and gravitational potential energy.
The escape velocity \( v_e \) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 2: Square the escape velocity equation.
To eliminate the square root, we square both sides:
\[ v_e^2 = \frac{2GM}{R} \]
Substituting the given escape velocity \( v_e = 100 \, \text{m/s} \):
\[ (100)^2 = \frac{2GM}{R} \]
This simplifies to:
\[ 10000 = \frac{2GM}{R} \]

### Step 3: Rearrange to find \( \frac{GM}{R} \).
From the equation \( 10000 = \frac{2GM}{R} \), we can rearrange it to find:
\[ \frac{GM}{R} = \frac{10000}{2} = 5000 \]

### Step 4: Use the gravitational potential energy formula.
The gravitational potential energy \( U \) of a mass \( m \) at a distance \( R \) from the center of a planet is given by:
\[ U = -\frac{GMm}{R} \]
Substituting \( \frac{GM}{R} = 5000 \) and \( m = 1 \, \text{kg} \):
\[ U = -5000 \times 1 = -5000 \, \text{J} \]

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

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