An atom bomb weighing 1 kg explodes relesing `9xx10^(13)` J of energy. What percentage of mass is converted into energy ?

Options

A. `0.1%`
B. `1%`
C. `2%`
D. `10%`

Explanation

To solve the problem of finding the percentage of mass converted into energy from an atom bomb explosion, we can follow these steps:

### Step 1: Identify the given data
- Mass of the atom bomb, \( m = 1 \, \text{kg} \)
- Energy released, \( E = 9 \times 10^{13} \, \text{J} \)

### Step 2: Use Einstein's mass-energy equivalence formula
Einstein's equation relates mass and energy:
\[ E = mc^2 \]
Where:
- \( E \) is the energy (in joules)
- \( m \) is the mass (in kilograms)
- \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \))

### Step 3: Rearrange the formula to find the mass converted to energy
We can rearrange the equation to find the mass equivalent of the energy released:
\[ m = \frac{E}{c^2} \]

### Step 4: Substitute the values into the equation
Substituting the values we have:
\[ m = \frac{9 \times 10^{13} \, \text{J}}{(3 \times 10^8 \, \text{m/s})^2} \]

### Step 5: Calculate \( c^2 \)
Calculating \( c^2 \):
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \]

### Step 6: Substitute \( c^2 \) back into the mass equation
Now substituting back:
\[ m = \frac{9 \times 10^{13}}{9 \times 10^{16}} = 10^{-3} \, \text{kg} \]

### Step 7: Calculate the percentage of mass converted to energy
Now we need to find what percentage of the original mass (1 kg) this converted mass (0.001 kg) represents:
\[ \text{Percentage} = \left( \frac{\Delta m}{m} \right) \times 100 = \left( \frac{10^{-3} \, \text{kg}}{1 \, \text{kg}} \right) \times 100 \]

### Step 8: Final calculation
Calculating the percentage:
\[ \text{Percentage} = 0.001 \times 100 = 0.1\% \]

### Conclusion
Thus, the percentage of mass converted into energy is **0.1%**.

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