We started by measuring our feet length and hip height (in meters, of course, since this is science). My feet are 0.35 m long and my hip height is 1.00 m. Then we calculated the (hip height / foot length) ratio for everyone and averaged all the values. Mine was 2.9 but, for the kids participating, it was around 3.2.
Turns out, from fossil data, that bipedal dinosaurs have an average (hip height / foot length) ratio of around 4.0. Why is this important? Because hip height is related to stride length, the average length of a dinosaur's steps. It also turns out that the height of an average bipedal dinosaur is 10 times their foot length. One footprint can now give us some useful information.
The next thing I did with the kids is set up a 25 meter long area in my backyard. Then the kids took turns walking this distance and then running this distance. In each case, they counted the number of footsteps and I timed them. From this, we calculated their walking and running stride lengths (footsteps / 2) and their walking and running velocities in meters per second.
When I did it, I got 27 steps in 19 seconds for walking (a stride length of 1.85 m and a velocity of 1.3 m/s) and 16.5 steps in 7.5 seconds for running (a stride length of 3.03 m and a velocity of 3.3 m/s). The kids, obviously got different values.
Turns out that studies have shown a roughly linear relationship between stride length and velocity for all different animals (the graph at left shows some mammals but these studies have been done for reptiles as well).
In other words, the longer the stride length, the faster the animal is moving (makes sense). It's also somewhat independent of the type of animal (not exactly true, but as a generalization it works well enough).
Next I had the kids calculate their gait for running and walking using data they had already obtained. The gait is defined as the (stride length / hip height). My gait for walking was 1.85 and for running was 3.03. Paleontologists use the generalization that a gait around 2.0 indicates walking and a gait around 3.0 indicates running. Once again, we compared numbers and discussed why they might be different.
Then we moved into the front yard. Before the kids came over, we had cut out two types of dinosaur tracks from construction paper - one was meant to represent a theropod (a bipedal meat-eating dinosaur) and one was meant to represent an ornithopod (a bipedal plant-eating dinosaur). The theropod had a walking gait to start and then a running gait. The nearby ornithopod track started with two tracks next to each other (standing) and then running. The scenario was that the theropod was walking along, saw the ornithopod, and took off running after it. The ornithopod, obviously, was running for its life.
Dinosaur tracks on my front lawn
When we moved into the front yard, I told the kids to imagine they were out in the hot Montana sun looking at a trackway eroding out of a sandstone bed. The kids figured out the scenario pretty quickly and then we did some measurements of foot length and stride length (two different stride lengths for the theropod and one for the ornithopod). From the foot length, we could estimate the dinosaur's hip height and overall height and from the stride length we could estimate their speeds.
My son sketching a dinosaur track
At this point it became more complicated and I left it for parents to go further with their kids if they wanted to do so. The formula paleontologists actually use to calculate a dinosaur's velocity is actually fairly complex:
v = 0.25 g0.5 SL1.67 L-1.17
Where g is the acceleration of gravity (9.8 m/s2), SL is the stride length in meters, and L is the hip height in meters. A little beyond 11-year-old kids, but I provided the information anyway.
For our dinosaur trackway, the data worked out to:
Theropod walking v = 0.25 (9.8 m/s2)0.5 (0.25 m)1.67 (0.72 m)-1.17 = 0.11 m/s
Theropod running v = 0.25 (9.8 m/s2)0.5 (0.85 m)1.67 (0.72 m)-1.17 = 0.88 m/s
Ornithopod running v = 0.25 (9.8 m/s2)0.5 (0.60 m)1.67 (0.52 m)-1.17 = 0.72 m/s
Are these reasonable speeds? I don't know but that's not really the point. The point was seeing that we can calculate such things from trackway data and we can also study living animals (in this case, all of us) to gain some insights into extinct ones.
I thought it was all pretty cool, if I do say so myself. The kids got to see how paleontologists can extract a lot of useful information about dinosaurs from a set of fossil tracks. They learned some new terms (bipedal, paleontology, ichnology). How to properly use a metric tape measure. How to record data (metric units). Different ways to indicate math operations (my handout had X / Y rather than X ¸ Y which is what they usually see). How to use a calculator. How to round off calculated numbers (informally). Averaging. Interpreting numerical results (what does this answer indicate?). How math is useful in interpreting something found in the real world (a dinosaur trackway). An educational afternoon in the sun while their peers were sitting in a math classroom doing worksheets.
If you want to learn more, here's the classic paper that started it all:
Alexander, R.M. 1976. Estimates of speeds of dinosaurs. Nature 261: 129-130.
Here's the exercise I put together if anyone's interested. It was the first time I did this so I would change a few things if we did it again.