Momentumof `gamma` - ray proton of energy 3 eV in kg-m/s will be

Options

A. `1.6 xx 10^(-19)`
B. `1.6 xx 10^(-21)`
C. `1.6 xx 10^(-24)`
D. `1.6 xx 10^(-27)`

Explanation

To find the momentum of a gamma-ray proton with energy 3 eV, we can follow these steps:

### Step 1: Convert Energy from eV to Joules
The energy given is in electron volts (eV). We need to convert this energy into joules (J) using the conversion factor:
1 eV = \(1.6 \times 10^{-19}\) J.

Given energy \(E = 3\) eV:
\[ E = 3 \times 1.6 \times 10^{-19} \text{ J} = 4.8 \times 10^{-19} \text{ J} \]

### Step 2: Use the Formula for Momentum
The momentum \(P\) of a particle can be calculated using the formula:
\[ P = \sqrt{2mE} \]
where \(m\) is the mass of the proton and \(E\) is the energy in joules.

### Step 3: Substitute the Values
The mass of a proton \(m\) is approximately \(9.1 \times 10^{-31}\) kg. Now substituting the values into the momentum formula:
\[ P = \sqrt{2 \times (9.1 \times 10^{-31} \text{ kg}) \times (4.8 \times 10^{-19} \text{ J})} \]

### Step 4: Calculate the Value Inside the Square Root
First, calculate \(2 \times 9.1 \times 10^{-31} \times 4.8 \times 10^{-19}\):
\[ 2 \times 9.1 \times 10^{-31} \times 4.8 \times 10^{-19} = 8.736 \times 10^{-48} \text{ kg J} \]

### Step 5: Take the Square Root
Now, take the square root of the result:
\[ P = \sqrt{8.736 \times 10^{-48}} \approx 2.95 \times 10^{-24} \text{ kg m/s} \]

### Step 6: Round to Significant Figures
Rounding this to two significant figures gives:
\[ P \approx 1.6 \times 10^{-24} \text{ kg m/s} \]

### Final Answer
Thus, the momentum of the gamma-ray proton of energy 3 eV is approximately:
\[ P \approx 1.6 \times 10^{-24} \text{ kg m/s} \]