A neutron moving with a speed u strikes a hydrogen atom in ground state in ground toward it with the same speed Find the minimum speed of the neutron for which ineleastic (completely or perially) collision may take place .The mass of neutron = mass of hydrogen `= 1.67 xx 10^(-27)kg`

Explanation

To solve the problem, we need to find the minimum speed of the neutron such that an inelastic collision with a hydrogen atom in its ground state can occur. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Energy Requirement for Inelastic Collision
In an inelastic collision, some kinetic energy is converted into internal energy of the system. For the hydrogen atom in its ground state, the energy required to excite it to the first excited state is approximately 10.2 eV.

### Step 2: Relate Kinetic Energy to the Energy Change
The kinetic energy (KE) of the neutron before the collision can be expressed as:
\[ KE = \frac{1}{2} m u^2 \]
where \( m \) is the mass of the neutron (or hydrogen atom, since they are equal) and \( u \) is the speed of the neutron.

### Step 3: Set Up the Condition for Inelastic Collision
For an inelastic collision to occur, the kinetic energy of the neutron must be greater than or equal to the energy required to excite the hydrogen atom:
\[ \frac{1}{2} m u^2 \geq \Delta E \]
where \( \Delta E \) is the energy required to excite the hydrogen atom to the first excited state (10.2 eV).

### Step 4: Convert Energy Units
To use consistent units, we need to convert 10.2 eV to joules. The conversion factor is:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
Thus,
\[ \Delta E = 10.2 \text{ eV} \times 1.6 \times 10^{-19} \text{ J/eV} = 1.632 \times 10^{-18} \text{ J} \]

### Step 5: Substitute and Solve for \( u \)
Now we can substitute \( \Delta E \) into the kinetic energy inequality:
\[ \frac{1}{2} m u^2 \geq 1.632 \times 10^{-18} \]
Substituting \( m = 1.67 \times 10^{-27} \text{ kg} \):
\[ \frac{1}{2} (1.67 \times 10^{-27}) u^2 \geq 1.632 \times 10^{-18} \]
\[ u^2 \geq \frac{2 \times 1.632 \times 10^{-18}}{1.67 \times 10^{-27}} \]
Calculating the right-hand side:
\[ u^2 \geq \frac{3.264 \times 10^{-18}}{1.67 \times 10^{-27}} \approx 1.95 \times 10^{9} \]
Taking the square root to find \( u \):
\[ u \geq \sqrt{1.95 \times 10^{9}} \approx 4.43 \times 10^{4} \text{ m/s} \]

### Step 6: Conclusion
Thus, the minimum speed of the neutron for which an inelastic collision may take place is approximately:
\[ u \approx 4.43 \times 10^{4} \text{ m/s} \]