D&D Physics Lab: Falling

At the start of the week we revisited the (to some) infamous dueling articles in The Dragon #88 by Parker and Winter: principally, should damage from falling be proportional (a linear function) to speed (and thus the square root of height), or instead to kinetic energy (and thus to height itself)? The physics theory seems clear: work is a linear function of energy, not speed. But we would not be scientists if we never put such things to the test.

Methodology

The experiment here is inspired by a teacher's guide for high school students: see Jones, "Understanding Car Crashes: It's Basic Physics", Insurance Institute for Highway Safety (2000). The goal of that experiment, "Lesson 2: Momentum Bashing" (p. 5-10) was to verify that momentum is a linear function of mass; here we modify it to check on the relation between fall height (or maybe speed) and work done.

The basic setup is to have some kind of track, one end raised on one or more books, with a ball rolling down into a catch-container. Jones' version used a marble, a ruler with a center groove, and a paper cup with a catch-hole cut in one side. I went looking for those things in my house and found I had none of them. So instead I scrounged up: a bouncy superball, a 15" cylindrical mailing tube, and an empty onion dip container with a hole cut out of it. Conveniently, the mailing tube was almost the same diameter as the ball; of course, if you want to try something similar, any replacement materials are fine.

For elevating one end of the track I used my stack of Conan paperback books. Fortunately, the books are all approximately the same width: 1/2" each. So where the track models the gravity of a falling body (angled tracks being a common means for "slowing down" a fall so we can observe it more closely), each added book will smoothly model an additional "floor" of the falling body. I ran the test ten times each with elevation heights of 1, 2, and 3 books. The distance that the catch-container gets pushed forward by the ball at the end is marked and measured with a millimeter ruler, and serves as a proxy for "work done" on the falling body at the end. See picture of apparatus below.

Results

Results of all testing are shown in the chart below, along with a linear regression on mean displacement of the catch-container at each height of fall. For all practical purposes, this proves to be a perfect correlation: the relationship is exactly D = 38h (where D is catch-container displacement in mm, and h height of fall in books), and the 3 data points match a straight line through the origin with R^2 = 99.9994% precision. I wish that any of my high school lab experiments had come out this accurately; I might have gone into a different profession!

Conclusions

Our experimental observations exactly match the physics theory; work done by a falling body is proportional to the height of the fall. In the old D&D falling debate, this again confirms Steve Winters' argument that "kinetic energy is the key", and that damage should indeed be a linear function of the fall height (not the speed, i.e., square root of height, as argued by Parker).