A 5 m ladder slides down a wall — how fast does the top fall?

The question

A 5 m ladder leans against a vertical wall. The base is pulled away from the wall at a constant 0.6 m/s. How fast is the top of the ladder sliding down when the base is 3 m from the wall?

What to notice first

The ladder never changes length, and that is the whole problem. A fixed length means the Pythagorean relation between the two distances holds at every instant, so differentiating it links the two rates. Nothing about the ladder's motion needs to be modelled beyond that one constraint.

Working it through

Call the base distance x and the wall height y. The ladder is the hypotenuse, and it is constant, so:

At the instant asked $about,x=3,soy$ follows from the same relation. This is a 3-4-5 triangle:

Differentiate the constraint with respect to time. The 25 is constant, so its derivative is zero — that is what couples the two rates:

Solve for the rate you want and substitute $x=3,y=4$ and $dx/dt=0.6\frac{m}{s}$:

The answer

The top slides down at 0.45 m/s. The minus sign is not decoration — it says the height is decreasing, which is the only thing distinguishing this from a ladder being raised.

The mistake this one catches

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