Algebra Lab. x = -16t 2 + vt + c. Using the Vertex Formula with Water Rockets!

Transcription

1 Algebra Lab x = -16t 2 + vt + c Using the Vertex Formula with Water Rockets!

2 Algebra Lab: Using the Vertex Formula with Water Rockets Objective: Students will apply the vertex formula to find the height of the flight path of a water bottle rocket. Students will use the quadratic that models the path of a projectile in feet/sec h = -16t 2 + vt + c, where v is velocity and c is starting height, t = time, and h = height. (Students should be familiar with this application of quadratics before attempting lab.) Materials Per Team: One 20 oz or 2 liter soda bottle (Do not use water bottles, they are not rated to withstand pressure.) A square of thin cardboard to make fins (about a foot by a foot) copy paper boxes and lids work well for this. An 8.5 x 11 piece of cardstock to make the nose cone A few strips of clear packing tape or duct tape. Scissors Cup Lab recording sheets. Materials for Class: Rocket launcher (directions on how to build on next page) Bicycle Pump with quality valve attachment for Schrader valves or a small air compressor with valve attachment. Source of water (bucket or drinking fountain is fine) Timer of some kind however, when I do this lab we just count from launch to landing, 1-Mississippi, 2-Mississippi, 3-Mississippi, etc A big, level, open area (football field), you don t need concrete, grass will work fine.

3 Building the Rocket Launcher: Materials: One ten food piece of ½ inch diameter PVC pipe. (You only need about seven feet, but they are sold in ten foot pieces) 4 end caps for ½ inch PVC pipe. 3 T connectors for ½ inch PVC pipe. 1 elbow with male and female end for ½ inch PVC pipe. 1 straight, short, metal, screw-on mount tire valve stem (available at automotive stores) Example: AutoZone part #1957 A drill or auger to drill the hole for the tire valve. One PVC pipe cutting tool or hack saw. Directions: 1. Cut 6 one-foot pieces of ½ inch diameter PCV pipe and 1 ten-inch piece. (Lengths don t have to be precise.) 2. Drill a hole in the center of one of the end caps large enough to Insert valve stem through. (You may want to sand the end of the cap down a bit so the top is flatter for a better seal.) Screw the valve stem into place. 3. Mark the ten inch piece of PVC pipe about 2 inches from the end. Hold the mark over a candle flame turning it for a few seconds until the PVC pipe is soft. Push the end of the pipe against something hard to create a slight bulge in the pipe. Cool under water to set the new shape.

4 3. Lay out the pieces as shown in the diagram below. End cap with valve (upright) T- connector End cap 10 inch piece (upright) T- connector Elbow (upright) End cap End cap T- connector 4. Glue the pieces together using PVC pipe glue. You will want to glue outside since the glue has strong fumes. Glue over a cardboard or drop cloth because the glue is drippy. Make sure to coat the outside of each pipe end in glue and the inside of each connector or end cap and twist a little when gluing to form a tight seal. (Be careful not to get glue in your tire valve.) 5. Important: Let dry over night! Do not try to launch the same day. Note: This is a very basic and safe launcher. There are designs on You Tube and on the internet for more advanced launchers including those with release valves that allow the pressure to build up higher and therefore shoot higher. Maybe someday I will be brave enough to try those out. But for now this fills the purpose and is still quite fun.

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6 Lesson Plan (break up as appropriate for your school s schedule): Part 1 Teach the formula for a projectile acting under the force of gravity in feet/sec.. This assumes you have already taught students how to find a vertex. h = -16t 2 + vt + c *** You may use the attached homework if you like. Use it before doing the lab. Part 2 Allow students to decorate their bottles with fins and nose cones using cardboard, cardstock, and packing tape or duct tape. Note: the fins must be placed evenly around the rocket or it will not fly straight up. Alternatively, rockets with no fins fly almost better, but it is more fun to deocorate. Part 3 Lead students through Step 1 on the lab handout. Part 4 Go outside, fill and launch rockets. Attach the bicycle pump to the rocket laucher. Have students fill their bottles about 1/3 of the way with water. One at a time have students come up and quickly turn their bottle over the upright pipe. (It s okay if some water goes into the pipe). They should twist the bottle a little and wedge it slightly on the bulge in the pipe. Then pump the bicycle pump until it launches. ***Do not let students use sandy bottles or water with dirt in it, it will break your launcher. *** Part 5 Come back inside and have students complete Steps 2 and 3 on their handout. They do not need to take the paper outside with them, they just need to remember how long it took until the rocket landed. Part 6 - Share out answers. Discuss the heights they found versus their estimation of the actual height of flight. Talk about the modeling of the formula and what other factors might have caused a difference between the answers we get from the formula and the actual height of the rockets.

7 Overlap and glue. Pattern for Nose Cone

8 Rocket Launching Lab Name Date Per Step 1: Begin with h = -16t 2 + vt + c and follow the steps to solve for velocity (v). *Remember this formula is for a projectile being launched straight up and then returning to Earth under the force of gravity as measured in feet per second. a. Your rocket will begin and end on the ground. Use the ground height (0) for the starting height (c) and the ending height (h). b. Add 16t 2 to both sides. Your new formula is: c. Divide both sides by t. This formula lets you find Step 2: Launch your rocket. Time the rocket from launch to landing in seconds. My rocket landed in seconds (t). Step 3: Use the formula from Step 1 to find out how high your rocket flew! a. Use the time from Step 2 for (t) in the formula you found in Step 1 to find the initial velocity of your rocket. The initial velocity of my rocket Work: is ft/sec b. Fill in the original formula h = -16t 2 + vt + c with the velocity (v) you found above and the ground height of zero as the starting height (c). Then use the vertex formula to find how many seconds (t) until your rocket was at maximum height, and substitute the time back in to find out how high (h) your rocket went. ***Do NOT use the (t) from step 2 (that is the time it took your rocket to land) You must solve for the time to vertex with the vertex formula. Work: According to my calculations, my rocket's maximum height was feet.

9 Answer Key: 1 second till landing = velocity of 16 and ½ second to vertex and vertex height of 4 feet. 2 seconds till landing = velocity of 32 and 1 second to vertex and vertex height of 16 feet. 3 seconds till landing = velocity of 48 and 1.5 second to vertex and vertex height of 36 feet. 4 seconds till landing = velocity of 64 and 2 second to vertex and vertex height of 64 feet. 5 seconds till landing = velocity of 80 and 2.5 second to vertex and vertex height of 100 feet. 6 seconds till landing = velocity of 96 and 3 second to vertex and vertex height of 144 feet. 7 seconds till landing = velocity of 112 and 3.5 second to vertex and vertex height of 196 feet. 8 seconds till landing = velocity of 128 and 4 second to vertex and vertex height of 256 feet. 9 seconds till landing = velocity of 144 and 4.5 second to vertex and vertex height of 324 feet. 10 seconds till landing = velocity of 160 and 5 second to vertex and vertex height of 400 feet.

10 Application of Quadratic Equations Name Date Per All of the following problems use a formula that models the path of a projectile up into the air and then falling back down under the force of gravity. 1 First, use the formula h = 16t 2 + vt + c (where h is highest point of arc, t is time, v is initial velocity, and c is starting height) 2 Then use the formula t b 2a to find the number of seconds it will take each projectile to reach its maximum height. 3 Finally, substitute your answer for t back in to the quadratic to predict what that height (h) will be. * Don t forget to answer with a sentence. 1. Diving Danny starts on a platform 50 feet above a pool. Assume his starting upward velocity (v) is 8 ft/sec. How many seconds until he reaches maximum height, and what will that height be? Sentence: 2. Colonel Cody is at a cliff top fort and fires a cannonball up into the sky hoping it will arc and hit a pirate ship out in the cove. If the cannonball is fired at an initial upward velocity of 224 ft/sec. from a 100 foot cliff, when will it reach maximum height and what will that height be? Sentence:

11 3. Slugger Sam hits a ball from homeplate. He smacks the ball at 4 feet off the ground at an initial upward velocity of 32 ft/sec. How many seconds until the ball reaches maximum height and what will that height be? Sentence: 4. Evil Kinevil rides his motorcycle up a 40 foot high launch ramp at an initial upward velocity of 128 ft/sec. The camera man wants to be able to count off the seconds after launch until Evil is at his highest point and have his camera aimed at the perfect spot. How many seconds should Evil tell him to count to, and what should he say his height will be? Sentence: 5. Samantha Spiker hits a volley ball for her team at an initial upward velocity of 16 ft/sec from a height of 5 feet. How long will it take for the ball to reach maximum height and what will that height be? Sentence: Could the ball clear a 7 foot net? 6. Domino the Dolphin s best jump into the air from the water is at an initial velocity of 64 feet per second. How many seconds until he reaches his maximum height and what will that height be? (Hint: Draw a picture to help you with this problem, it s tricky) Sentence: If the bottom of a 10 foot ring is hanging 60 feet over the water, could Domino s go through the ring?

12 Dear TpT Customer, I hope this lab is clearly written and makes sense to you. I do my best to double and triple check everything, however a mistake is always possible. If you need to contact me for any reason about this product, my direct is: diana.gillam@leusd.k12.ca.us Please visit my store on Teachers Pay Teachers for more helpful and time saving resources for Algebra 1. If you need more resources to teach quadratics, please check out my Unit 10 guided notes packet available in my store. And please don t forget to leave feedback if you liked this product. Sincerely, Diana Gillam (Algebra4All)

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