Tire
You have a hole in your tire and you watch it deflate. Draw the graph of tire pressure P against time t. What kind of curve is this? Find its equation.
This activity gets us down to basics. The air in the tire consists of a large number of molecules, which are constantly in motion. Now what happens when one of those molecules hits the inside surface of the tire? Well, it bounces off. And in fact that’s what causes the pressure in the tire--when you try to push the tire in with your finger it resists and that’s because of all those molecules colliding with the inside surface of the tire and pushing it back.
Now what happens if you punch a hole in the tire? Well then, a molecule heading for the inside surface of the tire might hit the hole instead, and shoot out and escape. Because of that the pressure in the tire will decrease over time.
Surprisingly enough, that’s all there is to it. Every molecule that flows out is one that was heading for the surface of the tire, minding its own molecular business, and found a hole instead (and tasted freedom!).
Surprisingly enough, that elementary observation is powerful enough to lead us to an equation of the pressure in the tire over time. You have a hole in your tire. You start with a tire at pressure 𝑃𝑃 = 400 kilopascals (kPa) and punch a hole in it. Over the next 60 minutes it goes down till the tire is quite flat. Explain why the graph of tire pressure 𝑃 against time t should have a concave-up shape.
Now suppose I tell you that over the first minute the pressure drops to 384 kPa –– a loss of 16 kPa. From that information, can you construct a formula for 𝑃(𝑡)?
