Speed III- the Experiment

Fossil Footprints----How Fast Is That Dinosaur Moving? Randall Caton (Christopher Newport University, Newport News, VA) Charlotte Otts (New Mexico State University, Grants Branch, Grants, NM) Retrieved on line and modified from: http://www.pcs.cnu.edu/~rcaton/dinorevweb.doc

Watch the motion video of the Velociraptor at: http://dsc.discovery.com/convergence/dinosaurplanet/interactive/dinoviewer.html

HOW FAST WAS THAT DINOSAUR RUNNING?

Type your guess here (in m/s):______

Because dinosaurs are vertebrates (like humans) we can learn much about dinosaur speed by studying human speed. That will be the goal of today's lesson. Since the most abundant information about dinosaurs comes from fossil footprints, let's begin our investigation by examining human footprints as they relate to other measurements.

We begin by gathering some data from our class to see if any patterns exist between footprint, leg length, and speed. For each member of the class, measure footprint (adjusting for the shoes of they are not removed), leg length (from the hip joint to the ground – with the shoe on), stride length (measuring from right heel to right heel in a normal step) and time in seconds required to walk 10 meters taking normal steps. Speed can be determined by dividing the distance (10 m) by the time (seconds) required to travel this distance. Complete the table below with your class data. Table 1: Student Footprint Leg Length (m) Stride length Speed (m/s) (m) (m) A B C D E F G H I

Some adjustments must be made. First, note that the faster one is going the longer their stride. However, size also affects stride length. We need to normalize for size and this is done by using: Relative stride length = (stride length)/(leg length) Measurements of a large number of modern organisms either walking, jogging, or running shows that this dimensionless stride length (relative stride length) is correlated to their “dimensionless speed” which is actual speed scaled to body size. By analyzing an abundance of data, the following formulas were established:

speed (m/s) = dim. speed x (leg length x gravitational acceleration)1/2 or dimensionless speed = speed (m/s) / (leg length x gravitational acceleration)1/2

We will use our calculators later to determine these measures.

But why did we need to measure the footprint? To determine the speed of a dinosaur we can not always measure the leg length – but we can measure the footprint from fossil remains, and in many cases can determine actual stride length based on multiple fossils at a single location. Because dinosaur body proportions are similar to that of humans, we can use relationships between human leg length and footprint, to predict dinosaur leg length from footprints. But first, we need some information about dinosaur footprints.

You have been given some photos of dinosaur footprints, but they are of course not actual size. Given the data on actual size of the footprint, use a proportion to determine the actual stride length of the dinosaur and record your values in the table.

Dino name Footprint Stride Actual Footprint Actual stride (photo) (photo) Velociraptor Apatopus Lineatus

Now, let's begin with the data analysis steps. First, we enter the human data into the calculator to make predictions using the dinosaur data.

Using the following steps, enter the footprint, leg length, stride length, and speed of the class members from the table above into L1, L2, L3 and L4 correspondingly.

ENTER THE DATA INTO A LIST: STAT 1 ( for Edit) To clear existing data: use the arrow keys to put the cursor on L1 CLEAR ENTER Type the data from column one pressing ENTER after each number Repeat this process to enter the data from the remaining columns into L2, L3 and L4 ***After pressing ENTER for the last data value, press 2nd MODE (for QUIT) to exit the data entry screen

To see the graph of the footprint vs. leg length data

GRAPH 2nd Y= (to enter stat plot) 1 (to select PLOT 1) Arrow keys to put the cursor on ON ENTER Arrow keys to put the cursor on the type of plot you want (line graph) ENTER IF NECESSARY: Change the Xlist to match the location of your x-values - L1. Change the Ylist to match the location of your y-values - L2. ZOOM 9 – to GRAPH allowing the calculator to set the graphing window min and max values SET THE WINDOW (Viewing Rectangle) If you want to set your own minimum and maximum on your x and y axis you can press: WINDOW Type a number for Xmin ENTER Type a number for Xmax ENTER Type a number for Xscl that determines the interval between the marks on the x axis ENTER repeat for Ymin, Ymax and Yscl Xres - leave this set at 1

To determine the relationship between footprint and leg length

Regression In analyzing data it is helpful to establish an equation representing y as a function of x. After graphing the data using zoom 9: Press MODE, FLOAT (floating decimals) Press STAT, → (CALC), number corresponding the type of function desired (we will use linear) Your calculator displays values for coefficients Paste your equation into Y1 by pressing: y=, VARS, 5 (statistics), →→ (eq), 1 (RegEQ) GRAPH - Your curve should go APPROXIMATELY through your data points.

Trace To use the equation in predicting values, y=, up arrow to highlight PLOT1, enter (this Turns PLOT off) GRPAH TRACE Type the value for the dino footprint and then ENTER. The corresponding y value – the dino leg length - will be displayed. Note: The value you type for x using the TRACE function must be between xmin and xmax – you may adjust your window as needed

Record the values in the table below and calculate the relative stride length:

Dino name Actual Actual Stride Leg Length Relative Stride footprint (from above) (From calculator) (Stride/Leg (From above) length) Velociraptor Apatopus Lineatus

To determine the relationship between Relative stride length and Dimensionless Speed

We must first analyze the relationship between relative stride length (RSL) and dimensionless speed (DS) in humans then apply that to dinosaurs.

On your calculator, press: STAT, 1 (EDIT), → until you see L5 at the top of the screen. Clear any data in L5 if necessary. Use the up arrow until L5 is highlighted and press 2nd 3 ÷ 2nd 2 (you should see L3/L2 on the status line at the bottom of the screen) Enter You should see data fill the list. This is the RSL for each class member. Right arrow to see L6. Up arrow to highlight L6. This formula (described above) is more difficult to enter – be careful with parentheses – and use 2nd 2 for L2, 2nd 5 for L5. Type: (L5)/((L3X9.8)^.5) ENTER

You should see L6 fill with data. This is the DS for the members of the class Return to the graph step above, and change L1 to L5 (RSL) and L2 to L6 (dimensionless speed). Then repeat the plotting of the data using zoom 9, the regression, turning off the plot, and trace determine the dimensionless speed of the dinosaur. Enter your data in the chart below.

Dino name Leg Length Relative Stride Dimensionless Actual Speed: Your guess (Stride/Leg Speed DS X ((LL X 9.8)^.5) length) Velociraptor Apatopus Lineatus

To determine actual speed, use the formula: speed (m/s) = dim. speed x (leg length x gravitational acceleration)1/2

HOW CLOSE WERE YOU????

Extensions:

1) Given that a sauropod has a leg length of 3.0 meters and relative stride length of 2.6 meters, how fast would a sauropod walk?

2) Young animals, such as puppies, have large feet relative to body size. If you had been investigating tracks of a young dinosaur with large feet, would the speed calculated from the tracks be overestimated or underestimated? Explain why.

3) How precisely can you make measurements? Can you reduce the scatter in your data by improving your measuring techniques?

4) Explain how the regression line is used to make predictions. Can you give an example of another dinosaur characteristic that could be investigated using a comparison to humans and a regression analysis?

5) Just what do they mean by a least squares regression line anyway? TO check it out go to: www.explorelearning.com Select browse gizmos Select data analysis and probability under 6-8 mathematics Select analyzing and displaying data Select Lines of best fit using least squares – activity A Note: you can only access this site 5 minutes unless you register. You can register for a 30 day free trial, or is it very inexpensive to subscribe for a year. R. McNeill Alexander (1991, 1992, 1995) from the University of Leeds has studied the relationship between stride length and walking or running speed in a variety of extant bipedal and quadrupedal birds and mammals, including humans. He has found that within a given group of animals there is a direct relationship between stride length and speed (the longer the stride length, the greater the speed). We present data in Fig. 1 taken by middle students in the Science of Living Spaces (SLS) program at Christopher Newport University and data from the dog of one of the authors. Notice that the two types of animals follow different curves and that there is considerable scatter in the human data.

Alexander found that if he took gravity and animal size into account, he found the same relationship across different sizes and in both bipeds and quadrupeds of different groups. He compared relative stride length, or stride length taking leg length into account, with dimensionless speed, or speed taking both leg length and the gravitational acceleration at the Earth’s surface into account.

Dimensionless speed is used by naval architects and engineers to make meaningful comparisons between scale models and real ships during the design process. Alexander found a relationship between relative stride length and dimensionless speed in a variety of bipedal and quadrupedal birds and mammals. As relative stride length increases, dimensionless speed also increases. In Fig. 2 we show the same data from the Science of Living Spaces students and dog replotted as relative stride vs. dimensionless speed. Notice that both animals now follow the same curve.