Not exponential. This time.

Not exponential. This time.

Guys, there’s always an exponential lurking somewhere. When the rate is proportional to the amount, the amount decays exponentially: a capacitor discharging through a resistor, detergent washing out of your sweater, foul odor leaving the room when you open a window.

But not this time.

In the previous post we derived the velocity of a jet flowing out of a hole given the pressure and the liquid density. And instead of the usual “velocity is proportional to the quantity” we get a small correction: it’s proportional to the square root of the quantity. A tiny change — and the whole solution behaves differently. Now Achilles can finally reach the tortoise in a finite time.

It always fascinated me that exponential decay is kind of contradictory. It’s blazingly fast (geometric progression!), and at the same time infinitely slow — because it never reaches exactly zero.

Here, when the outflow isn’t limited by viscosity but by inertia, the exponential law turns into a parabola — and the liquid leaves the vessel in finite time.

In the picture you can see the full derivation.

I totally forgot this funny fact from university physics. Thank you, Nikita, for bringing up this question in our conversation!