Question

A fly with a mass of \(1.0 \times 10^{-4} \mathrm{kg}\) crawls across a table at a speed of \(2 \mathrm{mm} / \mathrm{s} .\) Compute the de Broglie wavelength of the fly and compare it with the size of a proton (about $\left.1 \mathrm{fm}, 1 \mathrm{fm}=10^{-15} \mathrm{m}\right)$.

Short answer

Answer: The de Broglie wavelength of the fly is approximately \(3.313 \times 10^{-12}\) times larger than the size of a proton.

Step-by-step solution

Convert the given information

Before we start, let's convert the given speed of the fly to meters per second: \(2 \mathrm{mm/s} = 2 \times 10^{-3} \mathrm{m/s}\).

Calculate the momentum of the fly

The momentum of the fly can be calculated as follows: $$p = mv$$ where \(m\) is the mass of the fly and \(v\) is its speed. Plugging in the values: $$p = (1.0 \times 10^{-4} \mathrm{kg})(2 \times 10^{-3} \mathrm{m/s}) = 2 \times 10^{-7} \mathrm{kg \cdot m/s}$$

Calculate the de Broglie wavelength

Now we can calculate the de Broglie wavelength using the formula: $$\lambda = \frac{h}{p}$$ where \(h = 6.626 \times 10^{-34} \mathrm{Js}\) is the Planck's constant. Plugging in the values: $$\lambda = \frac{6.626 \times 10^{-34} \mathrm{Js}}{2 \times 10^{-7} \mathrm{kg \cdot m/s}} = 3.313 \times 10^{-27} \mathrm{m}$$

Compare with the size of a proton

The size of a proton is about \(1 \mathrm{fm} = 10^{-15} \mathrm{m}\). Comparing the de Broglie wavelength of the fly with the size of a proton: $$\frac{\lambda_{fly}}{\lambda_{proton}} = \frac{3.313 \times 10^{-27} \mathrm{m}}{10^{-15} \mathrm{m}} = 3.313 \times 10^{-12}$$ The de Broglie wavelength of the fly is much larger than the size of a proton, specifically, about \(3.313 \times 10^{-12}\) times larger.