The Terminal Length of Body Hairs

Hairs on different parts of the body have different lengths, and even after you shave off the hairs on a body part, they seem to eventually cease to grow after a while of growth. How, then, would they know when to stop, if the final length should be dependent upon where they grow? Parmesh asks me this question, so I am attempting to answer here.

Let the average length of the hairs of a body part be L[m], and let’s try to display it as a function of the time t[s]. Let the time when you shave that body part t = 0, then L(0) = 0. Let the speed of hair growth on the body part be g [m/s], and the probability of hair falling out per unit time be p[/s]. Then, we will have the following differential equation:

dL/dt = g – pL

The solution to this is:

L = g/p – C*e^(–pt) (where C is a constant)

The condition L(0) = 0 determines the constant C, and we obtain:

L = (g/p) * (1 – e^(–pt))

So the average length everlastingly increases to approximate g/p, while the rate of the increase everlastingly decreases to 0. While your hairs are short, the amount of hair falling out per unit time pL is small, since L is small. So the hair appears to be growing rapidly at this point. However, as the hairs get longer, the amount of hair falling out per unit time pL also increases, as L is now larger. Thus, after a long time, the growth per unit time g and the amount of hair falling out per unit time pL virtually balance each other out, and it seems as if the hairs stopped growing.

Now that the terminal average hair length is given by g/p, if the probability of hair falling out is the same throughout your body, the difference in length between your head hair and pubic hair should be attributed to that in speed of hair growth. Or, is pubic hair more prone to fall out?