The fission type of warhead of some guided missiles is estimated to be equivalent to `30000` tons of `TNT`. If `3.5 xx10^(8) J` of energy is released by one tone of expolining `TNT`, how many fissions occur in the explosions of warhead? An energy of `200 MeV` is released by fission of one atom of `.^(235)U` .

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the total energy released by the warhead.
The energy released by one ton of TNT is given as \(3.5 \times 10^8 \, \text{J}\). For \(30,000\) tons of TNT, the total energy \(E\) can be calculated as:
\[ E = 30,000 \, \text{tons} \times 3.5 \times 10^8 \, \text{J/ton} \]

### Step 2: Convert the energy released by one fission of \(^{235}\text{U}\) from MeV to Joules.
The energy released by fission of one atom of \(^{235}\text{U}\) is given as \(200 \, \text{MeV}\). To convert this into Joules, we use the conversion factor \(1 \, \text{MeV} = 1.6 \times 10^{-13} \, \text{J}\):
\[ E_{\text{fission}} = 200 \, \text{MeV} \times 1.6 \times 10^{-13} \, \text{J/MeV} \]

### Step 3: Calculate the number of fissions that occur.
The number of fissions \(N\) can be calculated by dividing the total energy released by the energy released per fission:
\[ N = \frac{E}{E_{\text{fission}}} \]

### Step 4: Substitute the values and calculate.
1. Calculate the total energy \(E\):
\[ E = 30,000 \times 3.5 \times 10^8 = 1.05 \times 10^{12} \, \text{J} \]
2. Calculate the energy released per fission:
\[ E_{\text{fission}} = 200 \times 1.6 \times 10^{-13} = 3.2 \times 10^{-11} \, \text{J} \]
3. Now, calculate the number of fissions:
\[ N = \frac{1.05 \times 10^{12}}{3.2 \times 10^{-11}} \approx 3.28 \times 10^{23} \]

### Final Answer:
The number of fissions that occur in the explosions of the warhead is approximately \(3.28 \times 10^{23}\).

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