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

Step by step solution

01

Velocity as a Function of Stride Length

Given the formula \(v = 0.78 s^{1.67} h^{-1.17}\) and setting \(h = 2\), we substitute \(h\) into the equation to find velocity \(v\) as a function of stride length \(s\):\[v = 0.78 s^{1.67} (2)^{-1.17}.\]Simplifying, we have:\[v = 0.78 s^{1.67} \cdot 0.4454 \approx 0.3474 s^{1.67}.\]

02

Graphing Velocity vs Stride Length

With the function \(v = 0.3474 s^{1.67}\), plot \(v\) against \(s\), considering stride lengths \(s\) from 2 to 10 meters. This graph would show a curve demonstrating how velocity increases with an increase in stride length.

03

Analyze the Effect of Increasing Stride Length

As \(s\) increases, since \(s^{1.67}\) implies a rapid increase relative to \(s\), \(v\) will increase, indicating that long strides allow the animal to move faster. Practically, a longer stride means covering more distance in less time.

04

Calculate Velocity for Specific Stride and Hip Height

For a stride length of 3 meters and a hip height of 2 meters, substitute \(s = 3\) and \(h = 2\) into the original formula:\[v = 0.78 (3)^{1.67} (2)^{-1.17}.\]Calculate \(v\):\[v \approx 0.78 \cdot 5.196 \cdot 0.4454 \approx 1.802 \, \text{m/s}.\]

05

Velocity as a Function of Hip Height

Given the stride length \(s = 3\), express \(v\) as a function of \(h\):\[v = 0.78 \times 3^{1.67} \times h^{-1.17}.\]Simplifying with \(3^{1.67} \approx 5.196\):\[v \approx 4.0516 \cdot h^{-1.17}.\]

06

Graphing Velocity vs Hip Height

With \(v = 4.0516 \cdot h^{-1.17}\), plot velocity \(v\) against hip height \(h\), considering \(h\) values from 0.5 to 3 meters. This graph would illustrate a decreasing curve as \(h\) increases.

07

Analyze the Effect of Increasing Hip Height

As \(h\) increases, \(h^{-1.17}\) causes a decrease in \(v\), indicating that animals with higher hip heights move slower when stride length remains constant, likely due to increased energy or force needed to lift and move taller legs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Formula

Understanding the velocity formula is crucial in analyzing how fast dinosaurs and other animals move. The velocity formula developed by R. McNeill Alexander gives us a way to calculate speed based on two main factors: stride length and hip height. Mathematically, the formula is expressed as follows: \[ v = 0.78 s^{1.67} h^{-1.17} \]Here, \(v\) is the velocity in meters per second, \(s\) is the stride length in meters, and \(h\) is the hip height in meters.
This formula shows us that:

• The velocity increases as the stride length increases, raised to the power of 1.67.
• The velocity decreases as the hip height increases, due to the negative exponent 1.17.

This complex relationship between stride length, hip height, and velocity is foundational in predicting the speeds of animals using just their tracks.

Stride Length

Stride length is the distance covered in one step by a dinosaur or any running animal. It is often measured from successive footprints of the same foot. In the context of Alexander's formula, stride length plays a vital role in determining the velocity. Larger stride lengths generally indicate that the animal was moving faster. Given the specific form of the formula where the stride length influences velocity through a power of 1.67, we understand the impact is quite significant. For example:

• If stride length increases, the effect on the velocity is more than linear due to the exponent 1.67.
• Animals with longer stride lengths can cover more ground each step, contributing to higher speeds.

In practical terms, for every increment in stride length, the velocity increases rapidly, hence larger stride lengths are often linked to more agile or faster animals.

Hip Height

Hip height in dinosaurs can be estimated based on the size of their footprints. It roughly correlates with their leg length and influences how quickly an animal can move. In Alexander's formula, hip height is a critical variable and affects velocity inversely.The formula tells us that as hip height decreases, velocity increases due to the negative exponent of 1.17 in the formula:\[ h^{-1.17} \]This means:

• Shorter hip height allows for greater speed, given the same stride length.
• Taller animals might experience more resistance or require more energy to move quickly.

Therefore, understanding hip height is essential for assessing the mobility of animals from fossilized tracks.

Graphing Functions

Graphing functions is a powerful tool in visualizing and understanding relationships in mathematical formulas. In the analysis of dinosaur velocities, graphing can help illustrate how stride length and hip height independently affect speed.To graph the velocity as a function of stride length, we use:\[ v = 0.3474 s^{1.67} \]for hip height of 2 meters. By plotting this from strides of 2 to 10 meters, we can see how velocity increases significantly as stride length grows.
For the function relating velocity to hip height, it is expressed as:\[ v = 4.0516 \, h^{-1.17} \]with a stride length of 3 meters. Here, the plot would show a decline in velocity as hip height increases from 0.5 to 3 meters.These graphical representations help us comprehend the trade-offs and dependencies within the formula, providing insightful visualizations of how animals possibly moved.

Alexander's Formula

Alexander's formula is a groundbreaking calculation in paleobiology and biomechanics, developed to estimate an animal's speed through their tracks. The components of stride length and hip height offer a practical means of reconstructing movement patterns of ancient animals, using: \[ v = 0.78 s^{1.67} h^{-1.17} \]Its predictive power lies in the detailed structure, which:

• Allows scientists to make educated guesses on dinosaur velocity using fossilized footprints.
• Informs us about the possible physical capabilities and behaviors of extinct species.
• Bridges the gap between simple observations of fossils and dynamic interpretations of ancient life.

Alexander's formula remains an essential instrument in studying prehistoric life, integrating mathematics with paleontology for a clearer picture of the past.