TODO: Biology deferred at the user's request: the cancer-treatment account, clinical dose choices, treatment planning and clinical interpretation of the numerical model.

For the specified energy-loss model, multiply

d E / d x = - k / E

2 E

and integrate from the entrance:

\frac{d ( E ^{2} )}{d x} = - 2 k, E (x)^{2} = E_{0}^{2} - 2 k x .

The positive stopping power is

S = - d E / d x = k / E

. The model reaches zero kinetic energy at

R = \frac{E _{0}^{2}}{2 k} = \frac{20 0 ^{2}}{2 ( 770 )} cm = 25.97 cm .

At the entrance,

S (0) = \frac{770}{200} = 3.85 MeV c m^{- 1} .

Half a centimeter before the formal endpoint,

E^{2} = 2 k (0.5 cm) = 770 Me V^{2}

, so

E = 27.75 MeV, S = 27.75 MeV c m^{- 1} .

The signed derivatives are the negatives of the quoted stopping powers. This inverse-energy stopping-power model deposits energy more strongly toward the endpoint. Its divergent endpoint derivative is integrable, giving finite total deposited energy

E_{0}

; a real stopping curve includes finite energy spread, range straggling, scattering and a more accurate low-energy stopping law. These numbers are consequences of the supplied ideal equation.