Calculate the de Broglie wavelength of a 1 mg stone moving at 10 m/s.

Question

ScanSolve AI Tutor · Accepted Answer

To find the de Broglie wavelength, we use the formula:

\lambda = \frac{h}{mv}

where $\lambda$ is the de Broglie wavelength, $h$ is Planck's constant, $m$ is the mass of the object, and $v$ is its velocity.

Identify the given values and constants.
- Mass of the stone, $m = 1 \text{ mg}$
- Velocity of the stone, $v = 10 \text{ m/s}$
- Planck's constant, $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$
Convert the mass to SI units (kilograms).
$m = 1 \text{ mg} = 1 \times 10^{-3} \text{ g} = 1 \times 10^{-6} \text{ kg}$
Substitute the values into the de Broglie wavelength formula.
$\lambda = \frac{6.626 \times 10^{-34} \text{ J}\cdot\text{s}}{(1 \times 10^{-6} \text{ kg}) \times (10 \text{ m/s})}$
Calculate the denominator ( $mv$ ).
$mv = (1 \times 10^{-6} \text{ kg}) \times (10 \text{ m/s}) = 10 \times 10^{-6} \text{ kg}\cdot\text{m/s} = 1 \times 10^{-5} \text{ kg}\cdot\text{m/s}$
Calculate the wavelength.
$\lambda = \frac{6.626 \times 10^{-34} \text{ kg}\cdot\text{m}^2\text{/s}}{1 \times 10^{-5} \text{ kg}\cdot\text{m/s}}$ $\lambda = 6.626 \times 10^{-34 - (-5)} \text{ m}$ $\lambda = 6.626 \times 10^{-29} \text{ m}$
The de Broglie wavelength of the stone is:
$\text{6.626} \times 10^{-29} \text{ m}$

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