Height of a Zero Gravity Parabolic Flight
Math 1010 Intermediate Algebra Group Project

Have you ever wondered what it might feel like to float weightless in space? One way to try it out is to fly on a special aircraft that astronauts use to train for their trips to space. Both NASA and the Russian Space Agency have been flying these for years.  The way this is accomplished is to fly to a high altitude, drop down to gain speed, and then start a large parabolic path up in the sky.  For a time ranging from 10 to 20 seconds, along the top part of the parabolic flight, an environment simulating zero gravity is created within the plane.  This effect can cause some nausea in the participants, giving rise to the name “Vomit Comet”, the plane used by NASA for zero-G parabolic training flights.  Currently there is a private company that will sell you a zero-G ride, though it is a bit expensive.
This lab will have you take a look at the parabolic path to try to determine the maximum altitude the plane reaches. First, you will work with data given about the parabola to come up with a quadratic model for the flight.  Then you will work to find the maximum value of the model.  Now for the data:
	Height of a Zero-G Flight t Seconds After Starting a Parabolic Flight Path	
Time t in seconds	2	20	40	
Height h in feet	23645	32015	33715	

To find the quadratic model, you will be plugging the data into the model .  The data points given are just like x and y values, where the x value is the time t in seconds and the y value is the altitude h in feet.  Plug these into the model and you will get equations with a, b and c.

Part 1:  Write your 3 by 3 system of equations for a, b, and c.

h(2)    = a(2)2 + b(2) + c = 4a + 2b + c = 23645
h(20)   = a(20)2 + b(20) + c = 400a + 20b + c = 32015
h(40)    = a(40)2 + b(40) + c = 1600a + 40b + c = 33715

Part 2:  Solve this system.  Make sure to show your work.
Solve 1st for c:
c = 23645 – 4a – 2b
 Substitute c and solve for b:
400a + 20b + (23645 – 4a – 2b) = 32015
400a – 4a + 20b – 2b = 32015 – 23645
396a + 18b = 8370
 Divide the last equation by 18
22a + b = 465
b = 465 – 22a
Swap b and c in last equation
1600a + 40(465 – 22a) + (23645 – 4a – 2b) = 33715
1600a – 40(22)a – 4a – 2b = 33715 – 23645 – 18600
716a – 2b = -8530
 Put B back in again in for the last equation
716a – 2(465 – 22a) = -8530
716a + 44a = -8530 + 930
760a = -7600
a = -10
Plug and Chug to find the remaining
b = 465 – 22(-10) = 685
c = 23645 – 4(-10) – 2(685) = 22315
 Solution
A = -10
B= 685
C= 22315

Part 3:  Using your solutions to the system from part 2 to form your quadratic model of the data.
h=at^2+bt+c.  
h(t) = -10t2 + 685t + 22315

Part 4:  Find the maximum value of the quadratic function.  Make sure to show your work.
t = – 2b/a
t= -685/-20  = 34.25
Plug 34.25 in to find max height
Max Height = -10(34.25)2 + 685(34.25) + 22315 = 34,045.625 feet

Part 5:  Sketch the parabola.  Label the given data plus the maximum point.  A good way to start labeling your axes is to have the lower left point be (0, 20000)

Time in Seconds

MAX HEIGHT
34.25, 34045.625

Part 6:  Reflective Writing.
Did this project change the way you think about how math can be applied to the real world?  Write one paragraph stating what ideas changed and why.  If this project did not change the way you think, write how this project gave further evidence to support your existing opinion about applying math.  Be specific.

I’ve always known math is an essential skill that we use in our daily lives, from building bookshelves, to cooking with a recipe, or putting gas in your car. I believe that 100% of us need basic math skills, including Algebra, and Geometry. I use these skills daily either at work, or on home projects. However, most of the math I have been learning in 1010 has no real life use, especially in this day and age. We have calculators, computer programs, and many other avenues to get the answer to nearly every problem in our book as fast as you can input it. All that aside, I’ve come to the realization that math is nothing more than brain exercises, which is something we all benefit from, and can apply to other areas. I’ve given up on asking “when are we ever going to use this”, or “why do we need to know this”, because the answer for 99% of us is NEVER. We will never need to know the square root of -21, or how fast a boat is traveling in still water through calculations, because we can use the technology that we have and just call the boat driver and ask him how fast he is going.