Cart and Pulley | Dr. Loveless Curiosity Lab

Prediction and Controls

Cart B moves at a constant speed of 2 ft/sec. Choose a rope length \(L\), guess what the speed of cart A will be when it is 0.5 ft from point Q, then press play. Watch the speed graph below as well to explore the behavior of the speed of cart A.

Rope Length \(L\) — ft

Guess for Cart A when 0.5 ft from Q — ft/sec

Adjust \(L\), make a guess, then press play or check your guess.

Animation Time \(T\) 0.44 sec

Cart and Pulley Visual

The animation begins at \(T=0.44\) and stops when cart A is 0.5 ft from point Q.

Speed Graphs

Green shows the constant speed of cart B. Purple shows the changing speed of cart A. The window expands as time increases.

Given

Fixed Rope Length \(L_1+L_2=L\), where \(L\) is fixed.

Cart B Cart B moves at constant \(\frac{dx}{dt}=2\) ft/sec, where \(x\) is the distance from Q to cart B.

Geometry Each rope piece forms a right triangle with height \(12\) ft.

Want

Let \(y\) be the distance from Cart A to \(Q\). We want to study:

\(\displaystyle \frac{dy}{dt}=? \)

The key question is how the fixed rope length forces cart A to move when cart B moves at a constant speed.

Results

The fixed-length condition gives the governing equation:

\(\displaystyle \sqrt{x^2+144}+\sqrt{y^2+144}=L. \)

We can differentiate this equation with respect to \(t\) to get a relationship between \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).

That is the central idea in related rates: write the constraint equation first, then differentiate to connect the rates.

Where It Started

Hannah started with a Math 124 related-rates problem involving carts, a rope, and a pulley. The original problem asks students to connect changing distances using a fixed rope length.

At first, it might seem natural that if one cart moves at a constant speed, the other cart should also move smoothly. But the rope creates a geometric constraint, and that constraint can produce surprising motion.

Going Further

This is one example of how a fixed-length constraint can lead to interesting speed behavior. Similar constraints appear in many simple machines and engineering systems: sliding ladders, pistons and rods, pulleys, crane cables, linkages, and rotating arms.

In each case, one distance or angle may change at a steady rate, while another part of the system changes at a very different rate. The geometry creates a relationship between those rates, and calculus gives us a clean way to study that relationship.

Prediction and Controls

Where It Started

Going Further

Further Reading