Potential energy of a 3kg body at the surface of a planet is `-54 J`, then escape velocity will be :

Options

A. 18 m/s
B. 162 m/s
C. 36 m/s
D. 6 m/s

Answer

To find the escape velocity of a 3 kg body at the surface of a planet where the potential energy is given as -54 J, we can follow these steps:

### Step 1: Understand the relationship between potential energy and escape velocity
The gravitational potential energy (U) of a mass (m) at the surface of a planet can be expressed as:
\[ U = -\frac{G M m}{R} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( m \) is the mass of the object (3 kg in this case),
- \( R \) is the radius of the planet.

### Step 2: Set up the equation with the given potential energy
From the problem, we know:
\[ U = -54 \, J \]
Thus, we can write:
\[ -\frac{G M (3 \, kg)}{R} = -54 \]
This simplifies to:
\[ \frac{G M (3)}{R} = 54 \]

### Step 3: Rearrange to find \( \frac{G M}{R} \)
Dividing both sides by 3 gives:
\[ \frac{G M}{R} = \frac{54}{3} = 18 \]

### Step 4: Use the escape velocity formula
The escape velocity (\( v_{escape} \)) from the surface of a planet is given by:
\[ v_{escape} = \sqrt{\frac{2 G M}{R}} \]

### Step 5: Substitute \( \frac{G M}{R} \) into the escape velocity formula
From Step 3, we know:
\[ \frac{G M}{R} = 18 \]
Thus, substituting this into the escape velocity formula gives:
\[ v_{escape} = \sqrt{2 \times 18} = \sqrt{36} \]

### Step 6: Calculate the escape velocity
Calculating the square root:
\[ v_{escape} = 6 \, m/s \]

### Conclusion
The escape velocity of the 3 kg body at the surface of the planet is:
\[ \boxed{6 \, m/s} \]

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