M&m Catapult Project

Math Fun: Parabolic Flight Materials: Stopwatch, Yard stick, Scotch tape, pen/pencil Overview: You will use catapult to launch M&Ms into the air. Using a stopwatch, you will time how long the projectile is in the air. That time will then be used to find an equation to model the flight of the M&M. Then the catapult will be moved on top of a desk, and you will use your equation to estimate how far the projectile will travel. Points will be scored based on where your M&Ms land on a target that you place on the floor. You will only have four attempts once the catapult is placed on the desk. You will also use the equation you create, as well as what you know about free fall, to determine the speed of the M&M at different times during its motion. When finished, we should have an idea of what the graph of the M&Ms position would look like (the equation you wrote), and what the graph of the M&M speed would look like. Directions: 1) Place the catapult on the ground. Be careful that it does not change position during tests. Fire a few practice rounds, making sure to pull the catapult back the same distance each time. It is very important to make sure that your shots are as consistent as possible! 2) You will need one person to be the launcher and one person to be the spotter. Make sure you watch where the M&M lands the FIRST time. Jessica will time for you. Try a few practice runs to make sure that you can hit close to the same spot with each shot. • MAKE SURE TO LAUNCH THE M&Ms THE EXACT SAME WAY EVERY TIME. • Put a piece of tape on the exact mark where each M&M lands. •

3) Once you get a fairly consistent launch time and launch distance, record 3 official times and their corresponding 3 official distances. Record these values in the table below. Average your three times and distances for the final row. Fill in the times and distances for each trial: Distance1:

Time1:

Distance2:

Time2:

Distance3:

Time3:

Average Distance: Average Time: 4) To find the height your M&M flew, you must first find the average fall time. Keep in mind, your M&M was rising for half of its flight, then fell for the other half. Therefore, to find the time it was falling you must divide your average time by 2.

Average fall time(t): 5) Now, you will find the vertex of the parabola. The equation for freefall of an object is d = 1/2gt2 where d is the vertical distance traveled, g is the effect of gravity in meters/s2, and t is the time in the air. The effect of gravity, g, is 9.8 meters/s2, which is 980 centimeters/s2. Plug in gravity and the freefall time you found in step 4 to find the vertical distance your M&M traveled.

1 (980 )t 2 2 d = 490 t 2

d =

d= You now have the maximum vertical distance your M&M traveled, which you know is the y value of the vertex.

y of the vertex: 6) To find the x value of the vertex, simply divide your average *horizontal* distance by 2 (from step 3).

x of the vertex: 7) Using the vertex you calculated and a start point of (0,0) , use vertex form to find a value for a.

vertex form: y = a(x - h)2 + k

8) Write the final equation for the flight of your M&M:

y= 9) Measure from the floor to the top of a desk.

Desk height: 11) Translate your equation by the height of the desk.

y=

12) Finally, put your equation into a graphing calculator and find where the M&M will land.

Distance the M&M will travel: 13) Get Jessica and place the target where you expect your M&M to land. You will be given four attempts. Record your top THREE scores below and ignore the worst result. Part of your grade WILL be based on your points.

There are NO PRACTICE RUNS from the desk. Shot 1:

________

Shot 2:

________

Shot 3:

________

Now for the Calculus!

What is the velocity of the M&M when it reaches the vertex?

What is the velocity of the M&M when it hits the ground? ((remember, velocity shows direction, too)

Find the velocity of the M&M at two other times.

If we wanted to graph the velocity function with respect to time, v(t), what would it look like?

Finding velocity when you know position (the parabola you graphed earlier), is finding the slope of the tangent line to the graph at that point. We have been doing this with free fall during chapter 2. We will continue looking at this idea, which is called taking the derivative.

Draw the graph of the position function from step 11. d(t) =

Draw a tangent line at the vertex. What is the slope of that tangent line?

Draw tangent lines at each of the other 3 times you picked to find the velocity. What are the slopes of those lines?

If we drew a graph using time as x values and velocity as y-values, what would it look like?