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

Options

A. (a)`18ms^(-1)`
B. (b)`162 ms^(-1)`
C. (c)`36ms^(-1)`
D. (d)`6ms^(-1)`

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 gravitational force.
The potential energy (U) of a body at the surface of a planet is given by the formula:
\[ U = -\frac{GMm}{r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( m \) is the mass of the body,
- \( r \) is the radius of the planet.

### Step 2: Substitute the known values into the potential energy equation.
We know the potential energy \( U = -54 \, J \) and the mass of the body \( m = 3 \, kg \). Thus, we can write:
\[ -54 = -\frac{GM(3)}{r} \]

### Step 3: Simplify the equation.
Removing the negative signs from both sides gives:
\[ 54 = \frac{3GM}{r} \]
Now, we can express \( \frac{GM}{r} \):
\[ \frac{GM}{r} = \frac{54}{3} = 18 \]

### Step 4: Use the escape velocity formula.
The escape velocity \( v_e \) is given by the formula:
\[ v_e = \sqrt{\frac{2GM}{r}} \]

### Step 5: Substitute \( \frac{GM}{r} \) into the escape velocity formula.
From the previous step, we have \( \frac{GM}{r} = 18 \). Therefore, we can substitute this into the escape velocity formula:
\[ v_e = \sqrt{2 \times 18} \]
\[ v_e = \sqrt{36} \]

### Step 6: Calculate the escape velocity.
Calculating the square root gives:
\[ v_e = 6 \, m/s \]

### Conclusion:
The escape velocity of the body at the surface of the planet is \( 6 \, m/s \).

---