Alexander’s formula: One interesting problem in the study of dinosaurs is to determine from their tracks how fast they ran. The scientist R. McNeill Alexander developed a formula giving the velocity of any running animal in terms of its stride length and the height of its hip above the ground.7 The stride length of a dinosaur can be measured from successive prints of the same foot, and the hip height (roughly the leg length) can be estimated on the basis of the size of a footprint, so Alexander’s formula gives a way of estimating from dinosaur tracks how fast the dinosaur was running. See Figure 2.57. If the velocity v is measured in meters per second, and the stride length s and hip height h are measured in meters, then Alexander’s formula is
(For comparison, a length of 1 meter is 39.37 inches, and a velocity of 1 meter per second is about 2.2 miles per hour.)
a. First we study animals with varying stride lengths but all with a hip height of 2 meters (so h = 2). i. Find a formula for the velocity v as a function of the stride length s. ii. Make a graph of v versus s. Include stride lengths from 2 to 10 meters. iii. What happens to the velocity as the stride length increases? Explain your answer in practical terms. iv. Some dinosaur tracks show a stride length of 3 meters, and a scientist estimates that the hip height of the dinosaur was 2 meters. How fast was the dinosaur running?
b. Now we study animals with varying hip heights but all with a stride length of 3 meters (so s = 3).
i. Find a formula for the velocity v as a function of the hip height h.
ii. Make a graph of v versus h. Include hip heights from 0.5 to 3 meters.
iii. What happens to the velocity as the hip height increases? Explain your answer in practical terms.