It's spring and you just finished a calculus exam. You leave the exam room with your book tied to a rope 1 m long. When you reach the sidewalk, you toss the book straight out perpendicular to the walk into the mud. Then you proceed down the sidewalk holding the rope and making a trail in the mud with your book. The tractrix is the path you've made in the mud with your human-powered tractor. A rough figure is Figure 1.1.

Figure 11.1: Tractrix Sketch

Mathematically, the length of the rope remains 1 m. and it acts to pull the book tangent to the furrow in the mud. We will use standard (x,y) coordinates to express these two facts. If the book is at the point (x,y) and you are at the point (x+h,0) on the sidewalk, then the length of the rope is the hypotenuse of a right triangle with base h and height y, so the Pythagorean theorem says

Draw the triangle from the position of the book at (x,y) perpendicular to the x-axis at (x,0) and over to your position at (x+h,0). Verify that this tells us that

The slope of the furrow is and that must also be equal to the slope of the hypotenuse of the triangle you just drew in the previous exercise.

Show that the slope of the hypotenuse of your triangle is

Equating the two slopes gives us a differential equation for the furrow, a curve known as the tractrix,

Separate variables in the equation above for the tractrix and integrate both sides. (Use Mathematicaif you wish, but be warned that we need the range where y starts at 1 and decreases.)

Since x=0 at the start when y=1, the value of your constant of integration C is ???.

Generalize this derivation of the tractrix to a rope of length k, showing that

The formula for x as a function of y is not really what we would like in terms of the physical question. We would like to specify the point (x+h,0) and compute both x and y. The differential equation needs to be used directly for those sorts of computations.

---

Previous project Next project Close this window