Temperature Dynamics

Differential Equations · Axiom Academy

Watch a cup of coffee cool, then take the thermometer yourself — by the end you'll have found the equation behind every cooling curve. A cup of coffee always cools the same way Pour a 90°C cup of coffee into a 20°C room and it cools fast at first, then slower, then barely at all — never quite reaching room temperature, but getting close enough that you'd need a very patient thermometer to tell the difference. That changing cooling speed isn't random. It's a rule. Press play and watch the thermometer fall. The mercury drops quickly at first, then eases off the closer it gets to room temperature — track the running time and the gap that's left to close. Notice the gap shrinks fastest at the start, when the coffee is hottest, and slowest near the end, when it's almost at room temperature. Does the starting temperature change the cooling rate? Drag the slider to set the coffee's starting temperature, then read the cooling rate off the thermometer. Try a temperature close to room temperature, then one far above it — watch how the rate itself changes. A cup at 180°C (fresh off the stove) cools nearly 3× faster than one at 60°C — same room, same coffee, only the gap is different. Find the ratio that never changes Drag the same temperature slider and watch where the point lands on the graph of cooling rate versus temperature gap. However far you drag it, the point always sits on the same straight line through the origin — meaning the RATIO of rate to gap never changes.

This is the written version of the interactive lesson above. See the full Differential Equations course.