What speed must an electron have if its momentum is to be the same as that of an x-ray photon with a wavelength of 0.20 nm?

The momentum of a photon is given by the following relationship:

where
h is the Planck constant
is the photon wavelength

For the photon in our problem, , so its momentum is


The electron must have the same momentum of this photon, and its momentum is given by (in the non-relativistic approximation)

where
m is the electron mass
v is its speed

Re-arranging this formula, we can calculate the electron speed:


And this velocity is quite small compared to the speed of light (), so the non-relativistic approximation that we used for the electron's momentum is valid.

The independent variable would be the fertilizer as it is what the dependent variable (the growth of grass) depends on.