Many things respond to a sudden change smoothly: a motor coming up to speed, a heater warming up. They follow a curve with one key number.
Key idea
A first-order system with time constant $\tau$ moves toward its new value as $y(t) = y_{\text{final}}\,(1 - e^{-t/\tau})$. After $\tau$ it has covered about 63%; after $3\tau$, 95%; after $5\tau$, more than 99%.
Why it works
The rate of change is proportional to how far there is left to go. That gives exponential approach, the same curve as on the Math Ladder's exponential models. Engineers read $\tau$ off a test run to know how fast a part can respond.
Worked example: How far after 2τ?
Final speed 200 rpm.
- $1 - e^{-2} \approx 0.865$.
- $0.865 \times 200 \approx 173\ \text{rpm}$.
Answer: About 173 rpm.
Common mistake
Thinking the system arrives after one time constant. It is only 63% of the way; plan on about $5\tau$ to settle.