Height of a Zero Gravity Parabolic Flight
Math 1010 Intermediate Algebra Group Project Project Completed by:
Daniel Musser, Jonathan Plastow, and Courtney Stowe
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:
Time ​
t​
in
seconds
Height ​
h​
in feet
Height of a Zero-G Flight t Seconds After
Starting a Parabolic Flight Path
2
30,506
10
31,250
20
29,300
To find the quadratic model, you will be plugging the data into the model h = at2 + 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.
Equation 1: 30,506 = 4a + 2b + c
Equation 2: 31,250 = 100a + 10b + c
Equation 3: 29,300 = 400a + 20b + c
Part 2: Solve this system. Make sure to show your work.
We'll start by taking away equation 1 from equation 2
Now we will go through that same process, only with equations 2 & 3 instead.
We now have two new equations that do not have c in them. We will now multiply equation 5
by 8 and then multiply equation 6 by negative 10. Finally, we will add those equations
together to eliminate b, so that we can then solve for a.
Now if we divide both sides by 1440, we get…
Now we will use this value for a in equation number 6 to solve for b
Now that we have the value for b, we can use equation 1 to solve for c
This makes the solution set:
a = -16
b = 285
c = 30,000
When we input these into the first three equations and they all held their equalities,
making this a verified solution set.
Part 3: Using your solutions to the system from part 2 to form your quadratic model of the data.
f(x) = -16x² + 285x + 30,000
Part 4: Find the maximum value of the quadratic function. Make sure to show your work.
We will now find the x and y pair for the maximum value of our quadratic function by using
the formula
b²
( −b
,
c
−
(
2a
4a )) We will start by plugging in the values from our quadratic function into
maximum value point for x.
−b
to find the
2a ​
Now if I divide -285 by -32 we get 8.90625, which is the x value for the maximum point in
our parabola. We will now plug in the values from our quadratic function into
to find the maximum value point for y.
b²
c − ( 4a
) Our resulting x and y pair that come from plugging the values of our quadratic function
into the formula that was previously given can are therefore:
X = 8.90625
Y = 31,269.140625
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)
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.
The reflective writing portion for each student can be found on the pages that follow.
Reflective Writing - Daniel Musser
This has changed my view in regards to using fractions. Before, I preferred
converting to decimals right away, but I found that keeping things in fraction format
actually keeps your math more accurate until you reach a point where you need to
convert the number to decimals.
However, my opinion about the usefulness of mathematics remains the same. I
have never questioned the usefulness of higher mathematics for engineers and
scientists, and I've always thought that more basic math is useful for everyone,
regardless of occupation.
Reflective Writing - ​
Jonathan Plastow
This project was a very difficult one for me because I really struggle with Parabolas.
However it was insightful to see a real life application of this type of mathematics. I did not
really have any changes in my ideas of parabolas because I don’t like them. But I will say that is
was interesting to figure out with my group how to solve this type of equation through some
trial and error. This project gave me an open view of a carrier that you might use this type of
mathematics and thus led me to stay away from a carrier that would have this type of math
involved, such as physics or any kind of engineering. Were an electrical engineer may use a
parabola to find the high and low of an electrical frequency. I will say however, that without my
excellent group. I would have done terrible on this assignment so I want to express my
appreciation for the group I have and being able to work together on this assignment.