To find the de Broglie wavelength associated with a particle of mass \(10^{-6} \, \text{kg}\) moving with a velocity of \(10 \, \text{m/s}\), we can use the de Broglie wavelength formula:
\[
\lambda = \frac{h}{mv}
\]
where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{J s}\)),
- \(m\) is the mass of the particle,
- \(v\) is the velocity of the particle.
### Step-by-Step Solution:
1. **Identify the values:**
- Mass \(m = 10^{-6} \, \text{kg}\)
- Velocity \(v = 10 \, \text{m/s}\)
- Planck's constant \(h = 6.63 \times 10^{-34} \, \text{J s}\)
2. **Substitute the values into the formula:**
\[
\lambda = \frac{6.63 \times 10^{-34} \, \text{J s}}{(10^{-6} \, \text{kg})(10 \, \text{m/s})}
\]
3. **Calculate the denominator:**
\[
mv = (10^{-6} \, \text{kg})(10 \, \text{m/s}) = 10^{-5} \, \text{kg m/s}
\]
4. **Now substitute this back into the equation:**
\[
\lambda = \frac{6.63 \times 10^{-34}}{10^{-5}} \, \text{m}
\]
5. **Simplify the expression:**
\[
\lambda = 6.63 \times 10^{-34 + 5} \, \text{m} = 6.63 \times 10^{-29} \, \text{m}
\]
6. **Final Result:**
The de Broglie wavelength associated with the particle is:
\[
\lambda = 6.63 \times 10^{-29} \, \text{m}
\]
