This activity is intended to supplement Calculus, Chapter 2, Lesson 4.
Time Required: 40-45 minutes
In this activity, students will investigate sinusoidal functions, collect data from a swinging pendulum, model the data with a cosine function, and take the derivative to find the velocity and acceleration.
Topic: Model Sinusoidal Data
Write an equation for a sinusoidal function.
First and second derivative to determine velocity and acceleration
Teacher Preparation and Notes
This activity utilizes a CBR or CBR 2, which records the movement of the pendulum and transmits it to the TI-89.
Students will need to know that the derivative of position is velocity, and the derivative of velocity is acceleration. Students are also expected to know the derivative of trigonometric functions.
The activity is designed to be a student-centered discovery/review involving data collection. If a motion sensor is not available, then use the data from the lists provided.
In order to collect data using the Calculator-Based Ranger 2TM, use the Ranger program. The Ranger program should be in the CBR device. To download it onto your calculator, connect the two, press 2ndLINK, arrow over to RECEIVE, and press ENTER.
To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11691 and select time, distance, and velocity.
- Student Worksheet: Getting the Swing http://www.ck12.org/flexr/chapter/9727
- time.89l, distance.89l, velocity.89l
- Calculator-Based Ranger (CBR) 2TM or Vernier GO! TM Motion
- pendulum (a ball hanging from a string)
- meter stick
- stop watch
Part 1 – Warm-up
Students begin the activity by answering questions to help them visualize and review characteristics of sinusoidal functions.
Ask students what the derivative of y=cos(x) is. When the A=1, B=1, C=0, and the D is set at π2, the graph looks likes y=−sin(x).
- range is from -1 to 1
- amplitude A=1
- period T=2π
A makes the amplitude bigger
C produces a vertical shift
B changes the frequency
B=2πT; The inverse function graphed on 3.2 confirms this.
Part 2 – Collect & Analyze Data
Be sure to use a smooth ball so that the sonic pulse is reflected off the surface well.
The graphs and equations for the experiment will vary. The following equations correlate to the data in the given lists.