Height of a Zero Gravity Parabolic Flight

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, around $5000. 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 10 20

Height h in feet: 30,506 31,250 29,300

To find the quadratic model, you will be plugging the data into the model ℎ = at^2 + bt + c. 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.

{ a(2)^2 + b(2) + c = 30506

{ a(10)^2 + b(10) + c = 32150

{ a(20)^2 + b(20) + c = 29300

Part 2: Solve this system. Make sure to show your work.

4a +2b + c = 30506 -25(4a + 2b + c = 30506)

100a + 10b + c = 31250 -100a - 50b - 25c = -762650

400a + 20b +c = 29300 __100a + 10b + c = 31250_____

-40b - 24c = -731400

-100(4a + 2b + c = 30506) 9/2(-40b - 24c = -731400)

-400a + -200b - 100c = -3050600 180b + 108c = 3291300

___400a + 20b + c = 29300_________ __-180b - 99c = -3021300

-180b - 99c = -3021300 9c = 270000

c = 30000

-40b - 24c = -731400

-40b - 24(30000) = -731400 4a + 2(285) + 30000 = 30506

b = 285 4a + 570 + 30000 = 30506

4a = -64

a = -16

a = -16

b = 285

c = 30000

Part 3: Using your solutions to the system from part 2 to form your quadratic model of the data.

h(t) = -16t^2 + 285t + 30000

Part 4: Find the maximum value of the quadratic function. Make sure to show your work.

h(t) = -16t^2 + 285t + 30000 h = -285 / (2(-16)) = 8.91

= -16(x - 8.91)^2 + 79.35 k = 8.91(-285 / (2(-16))) = 79.35

Maximum Value: (8.91 , 79.35)

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, 25000)

___x y___ 0 = -16(x- 8.91)^2 + 79.35

11.14 0 4.96 = (x - 8.91)^2

7 21 x - 8.91 = +- 2.23

10 60.34 x = 11.14 x = 6.68

6.68 0

8 66.10

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.

This particular project did not realy change the way I thought about math in the real world. I've known, like, basically my whole life that people who work with space and engineering will need to use this kind of math. Though, I will admit, this seems far too simple to be used professionally. I mean, I could be completely wrong with that way of thinking, but it just seems like people who work with space exploration will need to use some much more advanced math and science rather than a few quadratic equations and a parabola.

#math #Fall2015