Related Rates: Sliding Ladder

A ladder of length L leans against a wall with its foot x from the wall and its top y up it, so x² + y² = L². Pull the foot out and the top slides down. Differentiate with respect to time, with L constant, and 2x·dx/dt + 2y·dy/dt = 0, so dy/dt = −(x/y)·dx/dt. Worked example: a 5 m ladder with its foot 3 m out, moving away at 0.5 m/s. The top is at y = √(25 − 9) = 4 m, so dy/dt = −(3/4) × 0.5 = −0.375 m/s, the minus sign saying the top is going down. By the time the foot reaches 4.8 m the top is at 1.4 m and falling at 1.714 m/s, more than three times the foot's speed, though the foot never changed pace.

The classic mistake is putting x = 3 into the equation before differentiating. That freezes x into a constant, and its derivative is then zero whatever the ladder is doing; differentiate first, substitute the instant's numbers last. The second slip is the sign: the two speeds always have opposite signs, one end coming in while the other goes out. The formula's speed grows without limit as y approaches zero, which a real ladder never reaches, because it loses contact with the wall first. The same geometry is a working mechanism: the trammel of Archimedes slides a bar with its ends in two perpendicular grooves, and a pencil fixed between the middle of the bar and one end draws an ellipse.