Catapult Write up
The goals of the catapult project is to gather information about a catapult created by Ms. Lundin, and then use the equation to predict a landing point when launched from a different point. To complete this task we needed a catapult, ruler, a timer, a pack of
Lemonheads, for extra trials, and a data-keeping sheet. To start our catapult project we set the catapult on the floor. We had three team members, one team member was in charge of taking Data of the distance the Lemonhead flew and the time it was in the air
(Carly). The other was in charge of measuring how far the candy ball went from the starting point (Nick). I was in Charge of watching closely to where the lemonhead candy ball landed to help Nick with his measurement. We launched the Catapult three times to make sure it was accurate.
Data table:
Time (seconds)
0.9
1.0
Distance (cm)
157 cm
178 cm
0.9 140 cm
In order to find the an equation for our Catapult we had to find our average fall time, average air time, maximum height, and ½ of the distance where the lemon hit. Our average airtime, which was 0.93, was divided by 2 to get half of the average time, 0.465 because when something that is thrown into the air goes up, it takes half the time to come down. Our maximum height was 105.95025. The formula to find the maximum height is height = 1/2 (force of gravity)*(fall time)
2 . ½ (980)(.465) 2
=105.95025.

Reasoning and proof
Finding original equation of catapult
Mathematical Steps
Y=a (x-h) 2 +k
Y=a (x-79.166665) 2 +105.95025
Explanation h,k is the vertex of a parabola. This tells us we need to plug in half of the distance and the max height. H is the x coordinate of the vertex, and K is the y coordinate.
0=a (0-79.16665) 2 +105.95025
0 is the start point so we had to include 0 for x and 0 for y, in the equation to solve for a.
0=a*6267.358472+105.9502
-105.9502=6267.358472a
A=-0.0169054476
Table & Data from Our ground Launch
Y= (-0.0169054476) x-52.975125) 2 +105.9502
A is negative because parabola goes down.

Launch from Ground
120
100
80
60
40
20
-20
0
0 50 100
Horizontal Position
150
(x)Horizontal
Distance
(y)Vertical
Height
0 -2.42574E-07
39.583325
79.16665
118.749975
79.46437494
105.9525
79.46437494
158.3
0.089114965
Finding Translated Equation
Mathematical Steps
Y=a(x-h) 2 +k
Y= (-0.01690544)(x-79.16665) 2 +
105.9525105.9502
200
Explanation
To find the vertex we subtracted 20 from
79.16665 and added 73 to 105.9525. 73 cm was the height of the table, and we subtracted 20 because we were going 20 cm backwards. We translated so we could shift the graph, and because we are now launching the lemonade 20 cm behind the launch point we have to go back 20 cm because this time we launched it from the table which made the starting coordinates different.

Y=(-0.01690544) (x-59.16665) 2 +
178.9525
Launch from Table
(x)Horizontal
-20
19.583325
59.16665
98.749975
138.3
162.0524759
Predict Landing Point Using Quadratic Formula
Mathematical Steps
Y=(-0.01690544) (x-59.16665) 2 + 178.9525
Y=(-0.01690544) (x
Y=(-0.01690544)x 2
2
200
150
100
50
-100
0
0 100
-50
Horizontal Position
(y)Vertical Height
200
72.99999976
152.4643749
178.9525
152.4643749
73.08911496
-1.24613E-07
-118.33328x+3500.692) + 178.9525
+2.000477402x+119.7717269
Explanation
The first thing we need to do, is convert our vertex equation to standard form (Y=ax
2
+bx+c) so we can use the equation to predict our next landing point.
We squared (x-59.16665)
Next distribute (-0.01690544) to the three different terms inside the parenthesis.

QuickTime™ and a
de com press or are need ed to se e th is p icture.
This is the quadratic formula, which tells me my x intercept, in other words the landing point.
X=-43.71917588 and X=162.0524759
These were our answers from the quadratic formula. -43.71917588 is the x intercept behind the starting point, which doesn’t make sense because it is negative distant, and it is behind the starting point, and 162.0524759.
Connections:
During this project learning the quadratic, vertex, and standard form was important. I had to use what I learned from the quadratics in class in my problem to find the landing point. This project consisted of steps. I had to find the distance, and the height, and then go through the steps for translating the equation to vertex form. A quadratic function is anything that uses gravity. A quadratic function is a graph that has a parabola. In this project, the variables are height and horizontal distance. The distance the lemonhead goes determines height. The vertex equation tells me the max height and distance of the lemonhead. The standard form gave me the y intercept and the initial value from the original launch point, and the quadratic equation gave me the x intercept.
Errors that could have occurred while doing this project was getting the numbers to create the quadratic formula mixed up because I was absent for most of

the project. My quadratic equation had a lot of long numbers and decimals, which could have easily messed up my calculations when typing it in the calculator. It may have been hard to track the error so it took a close detail. If I had launched something else such as a gummy bear, it would have either went farther then the lemonhead I believe it would have had a shorter distance then the lemonhead. It also depends on the size of the lemonhead. Skills I used during this project was collaborating productively as well as solving problems resourcefully. It was hard to solve this equation on my own because of all of the class I had missed so I worked with Ms. Lundin after school to finish the equation. I worked close with my team to get the steps done on the equation, which helped me work better because the quadratic formula was confusing for me.
Other math that connects to the catapult problem is the Fireworks unit problem that I did in my 11 th grade year. In the Fireworks unit problem you had to take the same steps, finding the distance between the fireworks and the bridge and the height of how far up the fireworks go. The difference between these two equations was for catapult we had to create an equation using the information from launching the catapult, and fireworks the equation was given we just had to solve it.