Calculate the wavelength a particle of mass `m = 6.6 xx 10^(-27) kg` moving with kinetic energy `7.425 xx 10^(-13) J ( h = 6.6 xx 10^(-34) kg m^(2) s ^(-1)`

Explanation

To calculate the wavelength of a particle given its mass and kinetic energy, we can use the de Broglie wavelength formula. The formula can be expressed in terms of kinetic energy as follows:

\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

Where:
- \( \lambda \) is the wavelength,
- \( h \) is Planck's constant,
- \( m \) is the mass of the particle,
- \( KE \) is the kinetic energy of the particle.

### Step-by-step Solution:

1. **Identify the given values:**
- Mass \( m = 6.6 \times 10^{-27} \, \text{kg} \)
- Kinetic Energy \( KE = 7.425 \times 10^{-13} \, \text{J} \)
- Planck's constant \( h = 6.6 \times 10^{-34} \, \text{kg m}^2/\text{s} \)

2. **Substitute the values into the formula:**
\[ \lambda = \frac{6.6 \times 10^{-34}}{\sqrt{2 \cdot (6.6 \times 10^{-27}) \cdot (7.425 \times 10^{-13})}} \]

3. **Calculate the denominator:**
- First, calculate \( 2m \cdot KE \):
\[ 2m \cdot KE = 2 \cdot (6.6 \times 10^{-27}) \cdot (7.425 \times 10^{-13}) \]
\[ = 2 \cdot 6.6 \cdot 7.425 \times 10^{-27} \times 10^{-13} = 97.710 \times 10^{-40} = 9.7710 \times 10^{-39} \, \text{kg} \cdot \text{J} \]

4. **Take the square root of the result:**
\[ \sqrt{2m \cdot KE} = \sqrt{9.7710 \times 10^{-39}} \approx 3.12 \times 10^{-20} \, \text{kg m/s} \]

5. **Calculate the wavelength \( \lambda \):**
\[ \lambda = \frac{6.6 \times 10^{-34}}{3.12 \times 10^{-20}} \approx 2.11 \times 10^{-14} \, \text{m} \]

### Final Result:
The wavelength \( \lambda \) of the particle is approximately \( 2.11 \times 10^{-14} \, \text{m} \).

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