E=hf, Energy = Planck’s constant × frequency. (Planck’s constant is approximately 6.63×10–34 Js)
c=fλ, Wave’s speed = frequency × wavelength.
E=mc², Energy = mass × speed of light². (The speed of light is approximately 3.00×108 ms-1)
Now for a fun derivation exercise…
Equate the first two equations, this gives us hf=mc².
Rearrange the second equation to get f=c/λ and substitute into the equation above to get hc/λ=mc².
Cancel the c’s and rearrange for λ to get λ=h/(mc).
Now we have an equation that relates any particle with it’s wavelength!
This is called the de Broglie wavelength and has uses in areas of physics such as wave particle duality (Read more about light acting as a particle here). However, not only can waves like light act as particles, particles act like waves too! Shown by the photoelectric effect, photons above a certain frequency, when shone on the surface of a metal, can remove electrons from the surface. More importantly to this equation, electrons can be diffracted and show interference effects.
Let’s now find the wavelength of the average L. africana female African elephant in a stampede. These elephants weigh from 2160 to 3232 kg and taking the mean gives us 2696 kg. African elephants can move at speeds of 24.9 km/h which is 6.92 m/s. So their de Broglie wavelength is 6.63×10–34 ÷ (2696 ×6.92) = 3.55×10–38 m.
I hope you found this derivation and the power to calculate the wavelength of anything or anyone fun! Now you can time your friends running 100 m and see what their wavelength was during that time, which is a very important skill to have 🙂 .
sources:
https://physics.info/constants/
https://en.m.wikipedia.org/wiki/Matter_wave
https://en.wikipedia.org/wiki/Photoelectric_effect
https://en.wikipedia.org/wiki/Electron_diffraction
https://en.wikipedia.org/wiki/African_elephant
https://en.wikipedia.org/wiki/Fastest_animals
Photo by caterina sanders on Unsplash
Mathematters 