Help with a Physics/Calculus Problem - 40 yard dash

[dcebb2001]

Help with a Physics/Calculus Problem -- 40 yard dash

I have a physics problem that I just cannot solve.

I am trying to determine the velocity and acceleration for an athlete who is running a 40 yard dash.

What I know are the distances and split times for each player.

For example, a certain Player A may run a 40 yard dash with the following split times:

Distance / Time

0 yards / 0 seconds
10 yards / 1.78 seconds
20 yards / 2.92 seconds
40 yards / 5.13 seconds

Now I know I can graph the distance vs. time and use a cubic function with an order of 3 to get a really nice smooth line which allows me to "predict" or forecast what the time would be for this player at a certain distance or vice versus.

I also know that this polynomial curve gives me an equation of:

f(x) = 7.7083333E-05*x^3-0.0055125*x^2+0.22541667*x-4.5536491E-18

The R-Squared value of course is 1.


Now, I want to be able to use this distance v. time graph to determine the VELOCITY for this run along this nice smooth polynomial line. I know that I can take the average of the distances between the splits to get velocity, but I want to be more precise than that. The reason being is that I want to determine the velocity for Player A at 45 yards. In other words I want to FORECAST what his velocity would be at a distance of 45 yards if it was given that the player kept running.

I also want to be able to determine what the player's maximum velocity was during the run.

Next, I want to determine the ACCELERATION level during the course of the run. Most players will show the largest amount of acceleration during the first 10-20 yards. I want to know when that occurs given the nice polynomial fit I have.

Finally, I want to determine how fast the player falls off after attaining his top velocity. This would measure the player's stamina. In other words, the longer you stay at top speed, the more stamina you have. The faster you drop off and begin to decelerate, the less stamina you have. We can use the velocity and/or acceleration curves derived to figure this out as well.


I would love to be able to find a way to do this for several players in excel. Since I know all of the distances and split times, is this possible? Can I get a smooth line for velocity and acceleration in order to make forecasts using a cubic fit?

PLEASE HELP ME!

 

[SteamKing]

If you've got your equation fit, then finding velocity and acceleration would just be calculating the first and second derivatives of the equation and evaluating same at given distances from the starting line. The tools of the calculus should enable you to locate the points at which velocity and acceleration are maximum.

 

[rude man]

Change f(x) to x(t). Then v = dx/dt and a = d2x/dt2, i.e do as steamking suggests.