Galileo’s RampMath IB Final Project

OverviewHow does an object’s speed change as it rolls down a ramp? You are about to use math and physics to describe it. The experiment involves you rolling an object down a ramp and measuring its position and speed at set time intervals. I will provide you with a stopwatch, a ramp, and an object to roll. The challenge is to complete the experiment using only the provided materials.

To describe the motion, you need to create a drawing, a graph, an equation, and an explanation.

Here are a few equations and definitions that will help you in your investigation:

d :distance (measured∈meters)v : velocity (¿ , speed ,measured∈meters per second ,m / s )

a : acceleration(measured∈meters per second per second , ms2

)

v=d2−d1t 2−t1

a=v2−v1t 2−t 1

GroupingYou will work in groups of 2 or 3 students. Not 4. Don’t even ask.

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MondayLearning goal: Students will understand position and velocity versus time graphs.

Task: Complete the position-time and velocity-time graphs for the scenario your group chooses. Vocabulary:

Position: distance from 0, measured in meters.Velocity: how fast an object is moving in the positive or negative direction, measured in meters per second (m/s). Also called speed.

Example #1: Your teacher is exercising by walking around the track at school. She walks at a constant velocity of 1.4 m/s. Graph the first 60 seconds of her walk.

Example #2: Your teacher is feeling fit, so she runs for 30 seconds, gets tired and rests for 10 seconds, then runs for 20 seconds. Graph the 60 second “exercise” your teacher did.

Your story:

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Monday ConclusionsAn object is… Position-Time Graph Velocity-Time Graphnot moving or at rest

Horizontal line Horizontal line

moving away from you at a constant velocity

Line with a positive slope Horizontal line

moving away from you at an increasing velocity

Parabola Line with a positive slope in the first quadrant

moving away from you at a decreasing velocity

Radical Line with a negative slope in the first quadrant

moving toward you at a constant velocity

Line with a negative slope Line with 0 slope

moving toward you at an increasing velocity

Vertically reflected parabola Line with a positive slope in the fourth quadrant

moving toward you at a decreasing velocity

Vertically reflected radical Line with a negative slope in the fourth quadrant

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TuesdayLearning goal: Students will understand how to draw a position-time graph based on data they collect.

Task: Conduct the experiment described below, collect data, and analyze the results.Procedure:

1. Get a meter stick, a battery, a whiteboard, 2 different color markers, and a math book.2. Listen to the metronome (http://www.metronomeonline.com/) your teacher has turned on. It is

clicking once every second.3. Create a small ramp with your whiteboard and a thin book. The battery should roll down the

entire ramp in 4-5 seconds. If the ramp is too steep, the battery will roll too fast for you to observe.4. Describe your ramp with words and a picture.

5. What do you think will happen to the speed of the battery as it rolls down the ramp? Will the speed increase, decrease, or stay the same? Why?

6. What shape graph do you expect if you plot on a distance-time graph? Look at the page titled Monday Conclusions to decide.

7. Hold the battery at the top of the ramp and release it.8. Place a mark on the whiteboard for every click of the metronome.9. Measure the distance from each mark back to the battery’s starting point. Put those measurements

in the table below.Time Interval

1 s 2 s 3 s 4 s 5 s

Trial 1

Trial 2

Trial 3

Average

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Tuesday ConclusionsPlot the Average row from your data table on this graph.Sketch a curve that connects all your data.

What parent function does this graph look like?

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WednesdayLearning goal: Students will explore the meaning of a position-time graph’s most important features.

Task: 1. Present the graphs from the end of Tuesday. Each group will be responsible for explaining one

part of their graph (chosen randomly).2. Talk with your team to decide what these parts of the graph represent & complete the table.

Feature What does it mean in the real world?vertex

y-intercept

x-intercept(s)

3. The pictured quadratic function is from data collected by Slow Steve and his partner, Bored Bob. What happened?

4. The equation to the function in #3 is f(x) = 0.05(x-3)2. The whole function is shown below.

The vertex is at ( , )

How is f(x) = 0.05(x-3)2 related to the vertex?

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Wednesday ConclusionsSlow Steve and Bored Bob decided to play around in class instead of do their final project (do they really want to repeat Math 1B? Sheesh!). Steve threw the battery up in the air and Bob counted the seconds until it hit the floor.

The graph of Steve’s toss is shown below.

The vertex is approximately ( , )

How is f(x) = -9.8(x-0.3)2+2.8 related to the vertex?

What real event the x-intercept represent?

What real event does the y-intercept represent?

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Thursday & FridayLearning goal: Students will learn to summarize results from an experiment in a meaningful way.

Task: Everyone will create and discuss a whiteboard presentation.

Materials:Whiteboard, up to 4 different color markers

Project RubricObjective Evidence ScoreMM1A1Function notation,function characteristics,parent functions, andtransformations

P. 5 Tuesday’s Conclusions. 10%

MM1A3Solve quadratic equations algebraically andgraphically

Wednesday Conclusions. 5%

MM2AHorizontal transformations of functionsVertex form of a quadratic function

P. 6 #3 and #4 as well as Wednesday Conclusions.You can recognize when the vertex of a function has been shifted to the right.

5%

SMP2Reason abstractly and quantitatively

Work on pp. 2-3 as well as p. 6.You understand how position-time and velocity-time graphs operate as well as can explain how the graph of a function can represent a real event.

50%

SMP4Model with mathematics

Whiteboard presentations.A reasonable and accurate model is presented clearly.

30%

SMP6Attend to precision

Data table on p. 4.Results were recorded with reasonable precision (significant figures are used intelligently) and accuracy.

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