Chapter 2.2, Problem 18.1E

Alexander’s Formula One interesting problem in the study of dinosaurs is to determine from their tracks how fast they ran. The scientist R. McNeil 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.

The stride length of 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 2.45.

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 $v=0.78s^{1.67}{h}^{-1.17}$.

(For comparison, a length of 1 meter is $39.37\text{inches,}$ and a velocity of 1 meter per second is about $2.2\text{miles}\text{per}\text{hour}$.)

First, we study animals with varying stride lengths, but all with a hip height of $2\text{meters}$ (so $h=2$)

i. Find the 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 $2\text{to}10\text{meters}$.

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 $3\text{meters}$, and a scientist estimates that the hip height of the dinosaur was $2\text{meters}$. How fast was the dinosaur running?