Sliding Ladder | Dr. Loveless Curiosity Lab

Explore the Sliding Ladder

The four visuals move from a standard related-rates problem to richer geometric questions: speed, falling motion, hidden curves, and paths created when the point itself moves.

Concept 1: Falling Top

The base moves right at a constant speed. Watch the top slide down and reveal its vertical-speed graph.

Time \(T\) T = 1.00

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Concept 2: Fixed Firefighter

A fixed point on the ladder traces a hidden curve.

Consider the purple dot on the ladder. What curve do you think that dot traces as the ladder falls?

Point position \(d\) d = 0.61

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Concept 3: Hidden Envelope

Many ladder positions wrap around a hidden astroid envelope.

Run the simulation to watch ladder positions appear over time. Then reveal the hidden astroid curve that the ladder positions are touching.

Time \(T\) T = 0.00

Line density \(k\) k = 8

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Concept 4: Climbing Firefighter

Choose a motion model and see how the path changes when the firefighter moves along the ladder.

Now let the firefighter move along the ladder while it falls. Choose a motion model and watch the path change.

Motion model

Time \(T\) T = 0.00

Initial location 0.50

Initial velocity 0.00

Acceleration 0.00

Here \(a=0\) means the firefighter moves with constant speed along the ladder.

Center location 0.50

Wave size 0.50

Cycles during fall 1.00

The wave size is automatically scaled so the firefighter stays on the ladder.

Loading climbing firefighter visual...

Given

Ladder Length A fixed length \(L\).

Base Position The bottom of the ladder is at \((x,0)\).

Top Position The top of the ladder is at \((0,y)\).

Base Speed The base moves right at constant speed \(dx/dt=s\).

Model

The fixed ladder length gives the main constraint:

\(x^2+y^2=L^2\)

Differentiating gives:

\(2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}=0\)

\(\dfrac{dy}{dt}=-\dfrac{x}{y}\dfrac{dx}{dt}\)

Fixed Point

If the point is distance \(d\) from the top of a ladder of length \(L\), then:

\(X=(L-d)\cos(t),\quad Y=d\sin(t)\)

\(\dfrac{x^2}{(L-d)^2}+\dfrac{y^2}{d^2}=1\)

Envelope

The family of ladder positions forms an astroid envelope:

\(x^{2/3}+y^{2/3}=L^{2/3}\)

\(x=L\cos^3(t),\quad y=L\sin^3(t)\)

Moving Point

If \((x(t),0)\) and \((0,y(t))\) are the ends of the ladder, then the location of the firefighter is given by:

\(X(t)=(1-q(t))x(t), Y(t)=q(t)y(t)\)

Different choices of \(q(t)\) produce different curves.

Where It Started

Marcus started with a sliding-ladder related-rates problem and built a story around a firefighter, a tree, and a ladder sliding as the base moves away.

The classic question is simple to state: if the base moves at a constant speed, how fast does the top of the ladder fall?

Going Further

The same fixed-length constraint leads to several beautiful extensions: angle rates, paths traced by points on the ladder, hidden curves, envelopes, and moving points.

These ideas connect to linkages, robotic arms, piston mechanisms, and constrained motion in engineering.

Explore the Sliding Ladder

Where It Started

Going Further

Further Reading