© 21st Century Math Projects

Project Title: Quadratic Catapult Standard Focus: Algebra, Patterns, & Functions, Time Range: 3-5 Days Measurement

Supplies: Marshmallows (or small projectile), Yard Sticks, Popsicle Sticks, Glue (maybe Hot Glue), Rubber Bands and Masking Tape

Topics of Focus: -

Solving Quadratic Equations

-

Projectile Motion

-

Finding the Vertex

-

Using Physics formulas

Benchmarks: Seeing Structure in Expressions

A-SSE

3a. Factor a quadratic expression to reveal the zeros of the function it defines.

Creating Equations

A-CED

1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Creating Equations

A-CED

2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Creating Equations

A-CED

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Reasoning with Equations and Inequalities

A-REI

4. Solve quadratic equations in one variable.

Interpreting Functions

F-IF

7a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

F-IF

8a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Interpreting Functions

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Procedures: A.) Students will complete “Weapons Test” to begin practicing using quadratic equations to model position functions. Students will determine which of the 6 functions has a path that goes over a castle wall. Determining the vertex is an important skill that they must practice. B.) Students will complete “Incoming” to use the Kinematic Projectile Motion Formulas. A reference sheet to describe the variables is included and careful explanation of these are suggested. A sample problem is helpful. Determining the height is particularly tricky because students will use half of the time measurements to “figure out how high the object must have been for it to have taken that long to fall”. Students will then begin modeling their projectile paths and will need to solve simple systems of equations in order to calculate the coefficients for the position functions. Then the project – The Quadratic Catapult C.) Students will sketch a design with a materials list for a prospective “Quadratic Catapult” to be approved by the teacher. A useful, simple design can be seen here: http://www.stormthecastle.com/catapult/popsiclestickcatapult.htm

“Each team will be given 25 popsicle sticks, 5 feet of masking tape, 5 rubber bands, 2 pieces of paper and bottle of glue (or 5 minutes of access to the hot glue gun if available). You must make a functional catapult and some sort of device that can hold a marshmallow or another small projectile.” D.) Students build and test their design. E.) Designate a “launch line” on the floor for students to launch from. You will need stopwatches and yard sticks or tape measures. Students will record their information on “Catapult Competition Data”. They will also complete “Calculations” and “Follow-Ups”. Page 10-14 can be a printed and copied as a project packet. F.) Students compete in the Quadratic Catapult competition. The farthest toss wins! G.) Optional** Students compile data and present their findings. H.) Optional*** Students will complete “Catapult Data Crunch” to have more practice with the math applications. This can be used as an alternate assessment. Students will be given fictitious data from another Catapult Contest and they must figure out who would win the competition. The catch is that the winner of this contest would be the catapult that launches its projectile the highest.

© 21st Century Math Projects

’ Throughout the history of weaponry, throwing things has always been very popular. Throwing things of course led to launching things. When a projectile is thrown or launched its motion is parabolic and can be modeled by a quadratic equation. A group of overzealous medieval reenactors have created mathematical equations to model the position of each projectile they are testing. They are trying to determine if the weapons would clear a castle wall that is 20 meters away and 7 meters high. For Reference:

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Use the mathematical models to determine how far and high each projectile would go, determine if it would clear the wall and sketch and label a graph for each weapon in the space provided.

y = -0.1(x-11)2 + 12.1

Bombard

y = -0.035(x-17)2 + 10.11

Bow y = -0.05x2 + 1.25x

Catapult y = -0.18x2 + 3.6x

Culverin y + 0.055x2 = 1.925x

Sling 1000y – 528x = -12x2

Spear

© 21st Century Math Projects

Use information about the vertex and roots to answer to complete the chart. ? Bombard

y = -0.1(x-11)2 + 12.1

Bow

y = -0.035(x-17)2 + 10.11

Catapult

y = -0.05x2 + 1.25x

Culverin

y = -0.18x2 + 3.6x

Sling

y + 0.055x2 = 1.925x

Spear

1000y – 528x = -12x2

?

?

Sketch and label paths of each of the weapon projectiles to see if they clear the wall.

© 21st Century Math Projects

!!! Physics, Physics, Physics! If a projectile is in motion it is necessary to apply principles of physics to mathematical formulas. When an object is launched, there are initial conditions that can be used to answer a series of questions like “how high did the object go?”, “how fast did the object move horizontally?”, “how fast did the object move vertically?’ and “how high did it go?” Since a projectile has horizontal motion element AND vertical motion element it requires a special set of mathematical formulas --- The Kinematic Equations. Gravity will of course affect the vertical motion whereas the horizontal motion will remain constant. While there are a number of Kinematic Equations, there are three that will be the focus of this task.

𝑥 = horizontal distance 𝑥0 = starting position 𝑦 = final height

: 𝑥 = 𝑣𝑥 𝑡 + 𝑥0

𝑦0 = initial height 𝑡 = time (in seconds)

1

: 𝑦 = 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 2 : 𝑣𝑦 = 𝑣𝑦0 + 𝑔𝑡

𝑣𝑥 = horizontal velocity 𝑣𝑦 = vertical velocity 𝑣𝑦0 = initial vertical velocity 𝑔 = acceleration due to gravity (-9.81 m/s or -32.19 ft/s ) 2

2

*Calculating Height Hint – When trying to calculate the height, it is useful to creatively use the formulas. Since you know that parabolic motion is symmetrical. Halfway on its path, the object will begin to descend. Using half of the time allows for a more simple calculation.

For example,

suppose an object was in the air for 10 seconds. By 1

using t=5 with the equation: 𝑦 = 2 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 it allows you to set 𝑣𝑦0 = 0 (because it is for this moment suspended in mid-air with no velocity) and 𝑦0 = 0 (to see the difference) to calculate the distance the object descends in free-fall (which is the height!). Therefore 𝑦 = 1 𝑚 (−9.81 𝑠2 ) (5 𝑠)2 2

= −122.65 𝑚.

Thus, the object was at a height of 122.65 meters at 5 seconds. © 21st Century Math Projects

Use the Kinematic Equations for Projectile Motion to answer the following problems.

You will need to rearrange formulas to calculate the required parts.

.

A baseball player makes a throw that travels 200 ft the through the air. The ball was in the air for 2.1 seconds. For this and similar problems, assume 𝑥0 = 0 and and 𝑦0 = 0. a. What was the horizontal velocity of the throw? Knowns (w/units) 𝑥=

HINT: Use: 𝑥 = 𝑣𝑥 𝑡 + 𝑥0

𝑥0 = 𝑦=

b. When was the ball at its maximum height?

𝑦0 = 𝑡= 𝑣𝑥 =

c. What was the maximum height of the throw?

𝑣𝑦 =

HINT: Use: 𝑦 = 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0

𝑣𝑦0 =

1 2

.

𝑔=

.

An Olympic long jumper leaps with an initial horizontal velocity of 6 m/s and an initial vertical velocity of 4.5 m/s. Knowns (w/units)

a. How long was the jumper in the air?

𝑥=

HINT: Use: 𝑣𝑦 = 𝑣𝑦0 + 𝑔𝑡 (how long

𝑥0 =

did it take the jumper to reach the ground from her highest point?)

𝑦= 𝑦0 = 𝑡=

b. What was the maximum height of the jump?

𝑣𝑥 = 𝑣𝑦 = 𝑣𝑦0 =

c. How far was the jump?

𝑔=

© 21st Century Math Projects

!

To model the path of a projectile in motion with a quadratic equation this can be done with three points (the roots and the vertex) and a system of equations. Suppose a quadratic function has roots at x = 0 (0, 0) and x = 10 (10, 0) and the vertex at the point (5, 8). The Quadratic Equation can be found using the standard form: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and thus: Substitute in the variables to get 3 equations

0 = a(0)2 + b(0) + c

Simplify the equations

And using elimination

So substituting back in a.

(if the origin is a point c cancels)

0 = 100a + 10b

0 = 100(-0.32) + 10b

0 = c so

-16 = -50a – 10b

0 = -32 + 10b

0 = 100a + 10b

-16 = 50a, thus a = -0.32

32 = 10b thus b = 3.2

0 = a(10)2 + b(10) + c 8 = a(5)2 + b(5) + c 8 = 25a + 5b

and the equation is 𝑦 = −0.32𝑥 2 + 3.2𝑥

.

A bottle rocket is launched into the air from the ground at an initial upward velocity of 80 feet per second. The rocket landed 50 feet from the a. How long was bottle rocket in the air? launch site. b. What was the maximum height of the bottle rocket? c. Write an equation that models the path of the bottle rocket with respect to distance and height. d. Sketch and label the model below.

© 21st Century Math Projects

Use the information from questions 4 & 5 to sketch and label models on the graph below,

.

A boulder is launched from a catapult and it lands in 2.2 seconds, 48 feet a. What was the maximum height of the boulder? away. b. What was the horizontal velocity of the boulder? c. Write an equation that models the path of the boulder with respect to distance and height?

.

A cannonball is launched and it reaches its maximum height in 0.8 seconds and a. What was the maximum height of the lands 60 feet away. cannonball? b. What was the horizontal velocity of the cannonball? c. Write an equation that models the path of the cannon ball with respect to distance and height?

© 21st Century Math Projects

In this contest, a team of one to two students will design and build a catapult out of popsicle sticks to compete in a Quadratic Catapult competition. The goal is to build a catapult that launches a marshmallow (or a similar object) the farthest. Honorable Mentions are awarded to the marshmallow with maximum height and the marshmallow with the fastest velocity. In the first part of the project, you will enter the design phase. You will research to determine the best design for your catapult. You will need to consider what types of supplies will be available to you and how you plan to assemble the design. After you have an approved design, you will build, test, redesign and compete!

: Each team will be given 25 popsicle sticks, 5 feet of masking tape, 5 rubber bands, 2 pieces of paper and bottle of glue (or 5 minutes of access to the hot glue gun if available). You must make a functional catapult and some sort of device that can hold a marshmallow or another small projectile.

: 𝑥 = horizontal distance 𝑥0 = starting position 𝑦 = final height

: 𝑥 = 𝑣𝑥 𝑡 + 𝑥0 1

2

: 𝑦 = 𝑔𝑡 + 𝑣𝑦0 𝑡 + 𝑦0 2

: 𝑣𝑦 = 𝑣𝑦0 + 𝑔𝑡

𝑦0 = initial height 𝑡 = time (in seconds) 𝑣𝑥 = horizontal velocity 𝑣𝑦 = vertical velocity 𝑣𝑦0 = initial vertical velocity 𝑔 = acceleration due to gravity (-9.81 m/s or -32.19 ft/s ) 2

2

© 21st Century Math Projects

In the space below sketch a model for your catapult from at least two different angles. Include measurements and desired supplies. Your design will need to be approved by your teacher before you are permitted to build.

© 21st Century Math Projects

Name(s):

Test and measure 1. Position your catapult in the launch zone. 2. Launch the projectile and mark where it hits the ground. Measure the distance between the landing spot and the launch zone. 3. Record the time from the stopwatch from the moment it is launched to the moment it hits the ground. If there is an error, it will not count and it must be done again. 4. Repeat until you have three reliable trials. (Do not forget units!)

Time in the Air (in sec)

Distance Traveled (in ft)

© 21st Century Math Projects

Use the information from the Competition Data to complete the calculations in the table and determine a quadratic equation that models the projectile motion. Graph and label the three trials in the space provided.

Time in the Air Distance Traveled Acceleration Due to Gravity Horizontal Velocity 𝑥 = 𝑣𝑥 𝑡 + 𝑥0

Maximum Height 1 𝑦 = 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 2

Quadratic Equation that models the path of the projectile with respect to distance and height

© 21st Century Math Projects

Answer the following questions in complete sentences.

1.

What was the distance and height consistent from trial to trial?

2.

Was the velocity of the catapult consistent from trial to trial?

3.

Is there a relationship between height and the distance that your projectiles traveled?

4.

What are potential sources of error in this project?

© 21st Century Math Projects

: Before conducting the experiment, answer the following prompt in paragraph form. What strategies will you use to create a catapult that will launch a projectile the farthest? Why do you think your design will allow this? Is there a scientific explanation for this? How can you determine a mathematical equation that will model the path of your projectile?

: After conducting the catapult competition, answer the following prompt in paragraph form. What were the results for your catapult? Were your design choices successful? Is there anything you could have improved? Is there a scientific explanation for this? How can you determine a mathematical equation that will model the path of your projectile?

© 21st Century Math Projects

Each team is responsible for compiling their data and creating a PowerPoint, an I-Movie, or another multimedia program. The presentation should focus on the project and the design process. The presentation should include: original design ideas, items used in the design and why these items were chosen, changes that were made in the design after testing, the data, and a reflective analysis of the project (what would you change? what did you learn?)

Multimedia Project Rubric: CATEGORY

3

2

1

Presentation

Smooth delivery.

Fairly smooth delivery.

Content

Covers topic in-depth with details and examples. Subject knowledge is excellent.

Includes essential Content is minimal OR knowledge about the there are factual errors. topic. Subject knowledge appears to be good.

Mechanics

No misspellings or grammatical errors.

Four or fewer misspellings and/or grammatical errors.

More than four errors in spelling or grammar.

Attractiveness

Makes excellent use of font, color, graphics, effects, etc. to enhance the presentation.

Makes good use of font, color, graphics, effects, etc. to enhance to presentation.

Makes unattractive use of font, color, graphics, effects, etc.

Requirements

All requirements are met All, but one or exceeded. requirements are met.

More than one requirement was not completely met.

Originality

Product shows a large amount of original thought. Ideas are creative and inventive.

Little evidence of original thinking.

Product shows some original thought. Work shows new ideas and insights.

Delivery not smooth.

© 21st Century Math Projects

Below are measurements from a recent Quadratic Catapult Competition. However, this competition was a little different. It was not a distance competition. The projectile that reached the maximum height won the competition. Your job is to calculate the data from the contest to determine the winner. You can analyze the equation to determine why the winner may have won. 1.

From the data who do you predict won the contest? Why do you think they won?

Student

Horizontal Distance Traveled (ft)

Time in the Air (s)

Acceleration Due to Gravity (ft / s2)

Logan

8.4 ft

0.37 s

-32.19 ft/s

2

Rashon

6.3 ft

0.29 s

-32.19 ft/s

2

Paige

5.9 ft

0.34 s

-32.19 ft/s

2

Marrehn

9.9 ft

0.28 s

-32.19 ft/s

2

Deja

7.9 ft

0.42 s

-32.19 ft/s

2

Maximum Height (ft)

2.

Whose catapult had their projectile reach the maximum height in this competition? Did this match your prediction? Based on their data why do you think they did?

3.

Whose projectile traveled the farthest? Are these the same?

© 21st Century Math Projects

4.

If a catapult were to reach 1 foot in the air in this competition, how long would the projectile have to stay in the air?

5.

Does the horizontal distance have any impact on the height of the projectile?

6.

Whose device created the least height? Without seeing device, could you make a prediction about what their device might look like or how it might function? Why do you think it would be like that?

7.

What advice would you give Marrehn to improve her device?

8.

Based on your observation, is there a relationship between time, horizontal distance or vertical height?

© 21st Century Math Projects

Thank you for being my Math Friend! If you liked this

21st Century Math Project You might like others.

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Math it Up. Boomdiggy. © 21st Century Math Projects

’ Throughout the history of weaponry, throwing things has always been very popular. Throwing things of course led to launching things. To make a long story short, projectiles are in motion. When a projectile is thrown its motion is parabolic and can be modeled by a quadratic equation. A group of overzealous medieval reenactors have created mathematical equations to model the position of each projectile they are testing. They are trying to determine if the weapons would clear a castle wall that is 20 meters away and 7 meters high. For Reference:

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘

𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Use the mathematical models to determine how far and high each projectile would go, determine if it would clear the wall and sketch and label a graph for each weapon in the space provided.

y = -0.1(x-11)2 + 12.1

Bombard

(11, 12.1)

X = 0 and 22

(17, 10.11)

X = 0 and 34

(25/2, 7.81)

X = 0 and 25

(10, 18)

X = 0 and 20

(35/2, 16.8)

X = 0 and 35

(22, 5.808)

X = 0 and 44

y = -0.035(x-17)2 + 10.11

Bow y = -0.05x2 + 1.25x

Catapult y = -0.18x2 + 3.6x

Culverin y + 0.055x2 = 1.925x

Sling 1000y – 528x = -12x2

Spear

Use information about the vertex and roots to answer to complete the chart. © 21st Century Math Projects

?

?

?

Bombard

y = -0.1(x-11)2 + 12.1

22 meters

12.1 meters

No

Bow

y = -0.035(x-17)2 + 10.11

34 meters

10.11 meters

Yes

Catapult

y = -0.05x2 + 1.25x

25 meters

7.81 meters

No

Culverin

y = -0.18x2 + 3.6x

20 meters

18 meters

No

Sling

y + 0.055x2 = 1.925x

35 meters

16.8 meters

Yes

Spear

1000y – 528x = -12x2

44 meters

5.808 meters

No

Sketch and label paths of each of the weapon projectiles to see if they clear the wall.

!!! © 21st Century Math Projects

Physics, Physics, Physics! If a projectile is in motion, it is necessary to apply principles of physics to mathematical formulas. When an object is launched, there are initial conditions that can be used to answer a series of questions like “how high did the object go?”, “how fast did the object move horizontally?”, “how fast did the object move vertically?’ and “how high did it go?” Since a projectile has horizontal motion element AND vertical motion element this requires a special set of mathematical formulas --- The Kinematic Equations. Gravity will of course affect the vertical motion whereas the horizontal motion will remain constant. While there are a number of Kinematic Equations, there are three that will be the focus of this task.

𝑥 = horizontal distance 𝑥0 = starting position 𝑦 = final height

: 𝑥 = 𝑣𝑥 𝑡 + 𝑥0

𝑦0 = initial height 𝑡 = time (in seconds)

1

: 𝑦 = 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 2

𝑣𝑥 = horizontal velocity 𝑣𝑦 = vertical velocity

: 𝑣𝑦 = 𝑣𝑦0 + 𝑔𝑡

𝑣𝑦0 = initial vertical velocity 𝑔 = acceleration due to gravity (-9.81 m/s or -32.19 ft/s ) 2

2

*Calculating Height Hint – When trying to calculate the height, it is useful to creatively use the formulas. Since you know that parabolic motion is symmetrical. Halfway on its path, the object will begin to descend. Using half of the time allows for a more simple calculation.

For example, suppose an object was in the air for 10 seconds. By using 1 t=5 with the equation: 𝑦 = 2 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 it allows you to set 𝑣𝑦0 = 0 (because it is for this moment suspended in mid-air with no velocity) and 𝑦0 = 0 (to see the difference) to calculate the distance the object descends in free-fall 1

𝑚

(which is the height!). Therefore 𝑦 = 2 (−9.81 𝑠2 ) (5 𝑠)2 = −122.65 𝑚. Thus, the object was at a height of 122.65 meters at 5 seconds.

Use the Kinematic Equations for Projectile Motion to answer the following problems.

You will need to rearrange formulas to calculate the required parts.

© 21st Century Math Projects

.

A baseball player makes a throw that travels 200 ft the through the air. The ball was in the air for 2.1 seconds. For this and similar problems, assume 𝑥0 = 0 and and 𝑦0 = 0. a. What was the horizontal velocity of the throw? Knowns (w/units) HINT: Use: 𝑥 = 𝑣𝑥 𝑡 + 𝑥0

𝑥 = 200

95 ft/s

𝑥0 = 0

b. When was the ball at its maximum height?

𝑦= 𝑦0 = 0

2.1/2 = 1.05 seconds

𝑡 = 2.1 𝑣𝑥 = 𝑣𝑦 =

c. What was the maximum height of the throw?

𝑣𝑦0 =

HINT: Use: 𝑦 = 𝑔𝑡 2 + 𝑣𝑦0 𝑡 + 𝑦0 .

1 2

𝑔 = 32.2 ft/s

2

.

1

𝑦 = (32.2)(1.05)2 =17.75 feet 2

An Olympic long jumper leaps with an initial horizontal velocity of 6 m/s and an initial vertical velocity of 4.5 m/s. Knowns (w/units)

a. How long was the jumper in the air?

𝑥=

HINT: Use: 𝑣𝑦 = 𝑣𝑦0 + 𝑔𝑡 (how long

𝑥0 =

did it take the jumper to reach the ground from her highest point?)

𝑦=

0 = 4.5 + (−9.8)𝑡 … t = 0.46 sec so twice that is 0.92 seconds

𝑦0 =

b. What was the maximum height of the jump?

𝑡= 𝑣𝑥 = 6 m/s

At 0.46 𝑦 = 12 (−9.8)(0.46)2 +

𝑣𝑦 =

(4.5)(0.46)

𝑣𝑦0 = 4.5 m/s

=1.03 meters in the air.

𝑔 = 9.8 m/s

c. How far was the jump?

2

𝑥 = (6)(0.92) = 5.52 𝑚𝑒𝑡𝑒𝑟𝑠

© 21st Century Math Projects

!

To model the path of a projectile in motion with a quadratic equation this can be done with three points (the roots and the vertex) and a system of equations. Suppose a quadratic function has roots at x = 0 (0, 0) and x = 10 (10, 0) and the vertex at the point (5, 8). The Quadratic Equation can be found using the standard form: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and thus: Substitute in the variables to get 3 equations

0 = a(0)2 + b(0) + c

Simplify the equations

And using elimination

So substituting back in a.

(if the origin is a point c cancels)

0 = 100a + 10b

0 = 100(-0.32) + 10b

0 = c so

-16 = -50a – 10b

0 = -32 + 10b

0 = 100a + 10b

-16 = 50a, thus a = -0.32

32 = 10b thus b = 3.2

0 = a(10)2 + b(10) + c 8 = a(5)2 + b(5) + c 8 = 25a + 5b

and the equation is 𝑦 = −0.32𝑥 2 + 3.2𝑥

.

A bottle rocket is launched into the air from the ground at an initial upward velocity of 80 feet per second. The rocket landed 50 feet from the a. How long was bottle rocket in the air? launch site. 1

a.) 𝑦 = 2 (−32.2)𝑡 2 + (80)𝑡 0 = (−16.1)𝑡 2 + (80)𝑡 = 𝑡((−16.1)𝑡 + (80))𝑠𝑜 𝑡 = 4.97 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 1

b) y = 2 (−32.2)(2.485)2 + (80)(2.485) = 99.37 feet is the maximum. c) Use (0,0), (50, 0) and (25, 99.37) leads to y = 0.159x2 + 7.95x

b. What was the maximum height of the bottle rocket? c. Write an equation that models the path of the bottle rocket with respect to distance and height. d. Sketch and label the model below.

© 21st Century Math Projects

Use the information from questions 4 & 5 to sketch and label models on the graph below,

.

A boulder is launched from a catapult and it lands in 2.2 seconds, 48 feet a. What was the maximum height of the boulder? away. 1

a.) 𝑦 = 2 (−32.2)(1.1)2 = 19.48 feet b.) 48/2.2 = 21.8 ft/s c.) (0,0), (48,0) 𝑎𝑛𝑑 (24, 19.48) -0.034x2 + 1.62x

.

b. What was the horizontal velocity of the boulder? c. Write an equation that models the path of the boulder with respect to distance and height?

A cannonball is launched and it reaches its maximum height in 0.8 seconds and a. What was the maximum height of the lands 60 feet away. cannonball? 1 a.) 𝑦 = 2 (−32.2)(0.8)2 = 10.30 feet b.) 60/1.6 = 37.5 ft/sec b. What was the horizontal velocity of the c.) (0,0), (60, 0)𝑎𝑛𝑑 (30, 10.30) cannonball? 2 -0.011x + .69x **NOTE rounding even slightly will c. Write an equation that models the path of the cannon ball with respect to distance and height? affect the look of the graphs.

© 21st Century Math Projects

Below are measurements from a recent Marshmallow Catapult Competition. However, this competition was a little different. It was not a distance competition. The marshmallow that reached the maximum height won the competition. Your job is to calculate the data from the contest to determine the winner. You can analyze the equation to determine why the winner may have won. 1.

2.

From the data who do you predict won the contest? Why do you think they won?

Student

Horizontal Distance Traveled (ft)

Time in the Air (s)

Acceleration Due to Gravity (ft / s2)

Maximum Height (ft)

Logan

8.4 ft

0.37 s

-32.19 ft/s

2

0.55 ft

Rashon

6.3 ft

0.29 s

-32.19 ft/s

2

0.29 ft

Paige

5.9 ft

0.34 s

-32.19 ft/s

2

0.47 ft

Marrehn

9.9 ft

0.28 s

-32.19 ft/s

2

0.32 ft

Deja

7.9 ft

0.42 s

-32.19 ft/s

2

0.71 ft

Whose catapult had their marshmallow reach the maximum height in this competition? Did this match your prediction? Based on their data why do you think they did?

Deja’s catapult reached the maximum height.

3.

Whose marshmallow traveled the farthest? Are these the same?

Marrehn’s traveled the farthest, but that did not mean it went the highest.

© 21st Century Math Projects

4.

If a catapult were to reach 1 foot in the air in this competition, how long would the marshmallow have to stay in the air?

1 = (16.1)𝑡 2 = .25 seconds to reach the halfway point so .50 seconds overall.

5.

Does the horizontal distance have any impact on the height of the marshmallow?

No, it is not even relevant in the equation.

6.

Whose device created the least height? Without seeing device, could you make a prediction about what their device might look like or how it might function? Why do you think it would be like that?

Rashon may have a device with a shorter catapult or a basket that does release the marshmallow at the most optimal angle.

7.

What advice would you give Marrehn to improve her device?

Change the way the marshmallow is released so it may be able to launch it more vertically as opposed to horizontally.

8.

Based on your observation, is there a relationship between time, horizontal distance or vertical height?

There is a direct relationship between time and vertical height. There is no relationship between horizontal distance and time or vertical height.

© 21st Century Math Projects