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 seconds2
20
40
Height h in feet23645
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
System of equations:
(1) 4a + 2b + c = 23645
(2) 400a + 20b + c = 32015
(3) 1600a + 40b + c = 33715
Part 2: Solve this system. Make sure to show your work.
Solve Eq. (1) for c:
c = 23645 – 4a – 2b
Substitute for c in Eq. (2) and solve for b:
400a + 20b + (23645 – 4a – 2b) = 32015
400a – 4a + 20b – 2b = 32015 – 23645
396a + 18b = 8370
Divide this last equation by 18 to simplify:
22a + b = 465
b = 465 – 22a
Substitute for b and c in Eq. (3)
1600a + 40(465 – 22a) + (23645 – 4a – 2b) = 33715
1600a – 40(22)a – 4a – 2b = 33715 – 23645 – 18600
716a – 2b = -8530
Substitute for b again in the last equation
716a – 2(465 – 22a) = -8530
716a + 44a = -8530 + 930
760a = -7600
a = -10
Back substitute to find b and c:
b = 465 – 22(-10) = 685
c = 23645 – 4(-10) – 2(685) = 22315
System 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(t) = -10t2 + 685t + 22315
Part 4: Find the maximum value of the quadratic function. Make sure to show your work.
Vertex of parabola located at t = – 2b/a = -2(685)/(-10) = 34.25 seconds
Maximum value = -10(34.25)2 + 685(34.25) + 22315 = 34,045.625
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)
See file for graph
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 seconds2
20
40
Height h in feet23645
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
System of equations:
(1) 4a + 2b + c = 23645
(2) 400a + 20b + c = 32015
(3) 1600a + 40b + c = 33715
Part 2: Solve this system. Make sure to show your work.
Solve Eq. (1) for c:
c = 23645 – 4a – 2b
Substitute for c in Eq. (2) and solve for b:
400a + 20b + (23645 – 4a – 2b) = 32015
400a – 4a + 20b – 2b = 32015 – 23645
396a + 18b = 8370
Divide this last equation by 18 to simplify:
22a + b = 465
b = 465 – 22a
Substitute for b and c in Eq. (3)
1600a + 40(465 – 22a) + (23645 – 4a – 2b) = 33715
1600a – 40(22)a – 4a – 2b = 33715 – 23645 – 18600
716a – 2b = -8530
Substitute for b again in the last equation
716a – 2(465 – 22a) = -8530
716a + 44a = -8530 + 930
760a = -7600
a = -10
Back substitute to find b and c:
b = 465 – 22(-10) = 685
c = 23645 – 4(-10) – 2(685) = 22315
System 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(t) = -10t2 + 685t + 22315
Part 4: Find the maximum value of the quadratic function. Make sure to show your work.
Vertex of parabola located at t = – 2b/a = -2(685)/(-10) = 34.25 seconds
Maximum value = -10(34.25)2 + 685(34.25) + 22315 = 34,045.625
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)
See file for graph
001.jpg | |
File Size: | 1459 kb |
File Type: | jpg |
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.
It did change my way I think about math. I don't realize how many things are formulated by numbers. If NASA was off on there calculations it would cause big problems. I did not think how much math actually goes into a flight. I thought it was a computer that just flew the plane. Everyday life we use basic math. I use it at work and didn't even realize it until I started thinking of it. There are so many real world problems that we use math to find solutions. My opinion of math has been more positive after thinking in depth about how math can be important.
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.
It did change my way I think about math. I don't realize how many things are formulated by numbers. If NASA was off on there calculations it would cause big problems. I did not think how much math actually goes into a flight. I thought it was a computer that just flew the plane. Everyday life we use basic math. I use it at work and didn't even realize it until I started thinking of it. There are so many real world problems that we use math to find solutions. My opinion of math has been more positive after thinking in depth about how math can be important.