Question

If your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take you to move 0.80 m (one step)? Will you notice diffraction effects as you walk through doorways?

Short answer

1. $v=1.1\times {10}^{-35}m/s$
   
2. $t=2.3\times {10}^{27}yr$
   

Step-by-step solution

Calculate the velocity using the wavelength

First, we plug our value for$\lambda$ into equation (1) and solve for p, so we get the momentum required for you to undergo diffraction:

$$\begin{gathered} p=\frac{h}{\lambda } \\ p=\frac{6.626\times {10}^{-34}}{1} \\ p=6.626\times {10}^{-34}kg.m/s \end{gathered}$$

We know that the nonrelativistic momentum is given by:$p=mv$

Solving for$v$ and plug our values for p and m, we get your speed:

$$\begin{gathered} v=\frac{p}{m} \\ v=\frac{6.626\times {10}^{-34}}{60} \\ v=1.1\times {10}^{-35}kg.m/s \end{gathered}$$

Calculate the time

The time it takes you to move a distance$x=0.8m$ with this speed is:

$$\begin{gathered} t=\frac{x}{v} \\ t=\frac{0.8}{1.1\times {10}^{-35}} \\ t=7.24\times {10}^{34}s \\ t=2.3\times {10}^{27}yr \end{gathered}$$

Of course, you will not notice diffraction effects as you walk through the doorway because your speed is a way higher than the speed calculated in part (a) and your wavelength is very small that does not show a significant wave nature.