Acceleration 1-D Motion for Calculus Students (90 Minutes)

Transcription

1 Acceleration 1-D Motion for Calculus Students (90 Minutes) Learning Goals: Using graphs and functions, the student will explore the various types of acceleration, as well as how acceleration relates to position, velocity, and speed. Prior Knowledge: Students should be familiar with the concept of 1-D motion, graphs of time versus position, and graphs of time versus velocity and speed. Students should also be able to take derivatives, find critical points, and use concavity to describe functions. NCDPI/AP Objectives: Competency Goal 2: The learner will use derivatives to solve problems. 2.03: Interpret the derivative as a function: translate between verbal and algebraic descriptions of equations involving derivatives. 2.05: Interpret the second derivative: identify the corresponding characteristics of the graphs of ƒ, ƒ', and ƒ". 2.06: Apply the derivative in graphing and modeling contexts: interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration. Materials Needed: One large toy-car or truck (big enough for the class to see) Masking tape Graphing calculators for each student (or at least one for each group) Worksheet copies for each student Whiteboard/markers Think-Pair-Share Warm-up: Write the following example on the board or overhead for students to begin as they walk in. Question: Find the inflection points and intervals of positive and negative concavity of the position function,. Use a graph to support your answer. Simply looking at the position graph, what do you think might be the significance of d (t)? Why? Answer: The position function d(t) has an inflection point at t = 6. At that instance, d(t) changes from concave down to concave up. This means that d (t) changes from negative to positive at t = 6. The significance of d (t) is that it gives us the acceleration of an object. The students should spend 5 minutes working on the problem alone. Then, give them a few minutes to discuss their answers with a designated partner. Finally, a pair should be chosen to present or read aloud their answers. The share aspect should include some discussion of d (t),

2 and it s likely that one of the students will guess that d (t) represents acceleration. This is the ideal bridge into the formal lesson. Instructional Activities: Lecture: Begin the lesson with a simple lecture on basic concepts that students will use for their group activities. The students will fill in the notes on their worksheet. Before writing out the punch-line of each statement, ask for a student to volunteer an answer for what to write. Acceleration is the rate at which velocity is changing. Thus, a(t) = v (t) = d (t). If velocity is measured in m/s, then acceleration would be in m/. The critical points of v(t) indicate possible changes in the sign of acceleration. Before moving on to a demonstration, write the following example on the board for students to complete as independent practice. Give the students a few minutes by themselves, and then ask for a volunteer to come to the board and explain their answer. Question: Find where the acceleration is changing if the velocity of an object is given by: v(t) = 4t 2. Answer: When t = (2/3), acceleration changes from negative to positive. Demonstration: Before beginning the group activities, perform a simple demonstration in front of the entire class. Inform the students that acceleration isn t as intuitive as they might think. For example, what does it mean for an object to have negative velocity and positive acceleration? Is the object actually speeding up? You ll do a demonstration of 5 various types of acceleration, and the students will model each of these soon. Place a straight piece of masking tape on the floor or on a table. Tell the students that this will be our line of motion. Ask for a student or two come up to help stop the cars. o To the best of your ability, roll the car at a constant speed in any direction; a(t) = 0 o Roll the car in the right direction so that it s speeding up; +v(t), +a(t) o Roll the car in the right direction so that it slows down immediately; +v(t), a(t) o Roll the car in the left direction so that it s speeding up; v(t), a(t) o Roll the car in the left direction so that it s slowing down immediately; v(t), +a(t) Group Activities, Part 1: Tell the students it s time to form groups so they can consider these types of acceleration more seriously. Following the lesson-plan, I ve included a page of slips to

3 be copied (according to the number of students) and then cut out. Hand one slip to each student, and then tell the students to find their fellow group members. The functions are listed below: Group 1: v(t) = 6t a(t) = 6 Group 2: v(t) = -8t a(t) = -8 Group 3: v(t) = a(t) = Group 4: v(t) = a(t) = Group 5: 8t v(t) = 8 a(t) = 0 The students should work in groups on Part 1 of their worksheet for about 15 minutes, while you circulate and answer questions. Group Activities, Part 2 Jigsaw: After the groups have finished, scramble the groups so that each new group has exactly one member from each possible function. Now, for the next 20 minutes, the students should go over their answers as a team and fill out the rest of the worksheet, except for the last practice section. Each member should have a turn explaining their previous group s work. Whole-Class Discussion: After the groups have finished for the second time, the class should come back together for a discussion and to redo the demonstration from the beginning of the lesson. On an overhead or on the board, have a blank copy of the table. Now, ask for a volunteer from each group to explain their function and to fill in the table. After they ve filled it in, have them use the toy-car to model the object s path; this should make the conjecture about the relationship between velocity, acceleration, and speed more concrete. Position Function Is d(t) concave up, down, or neither? Velocity sign Object s speed increasing? Up + Yes + Down _ Yes _ Up _ No + Acceleration sign

4 Down + No _ 8t Neither + No 0 Finally, ask for volunteers to answer the questions about concavity, acceleration, and speed at the end of the worksheet. If the velocity function is constant, then the acceleration function is zero. If the position function is concave up, then acceleration is positive, and if the position function is concave down, then acceleration is negative. For an object s speed to be increasing, then the velocity and acceleration functions must have the same signs they should both be positive or both be negative. Independent Practice: Students should work on the last problem of the worksheet on their own for the remainder of the period. Before the end of class, be sure to go over the answers. Question: The position function of an object is given by. In what intervals is velocity negative? When t is within (-, 1) and (1, ). In what intervals is acceleration negative? When t is within (1, ). In what intervals is the object s speed increasing? Using the notion that velocity and acceleration should have the same sign, the speed is increasing when t is within (1, Closure: Inform the students that they ve delved quite deeply into the concepts of acceleration, and that they ve seen how position, velocity, speed, and acceleration work together! They re now prepared to move on to other exciting applications of derivatives, and eventually 2-D kinematics. Homework: The worksheet or textbook problems of your choice!

5 1-D Motion - Acceleration Name: Student Notesheet Warm-up: Find the inflection points and intervals of positive and negative concavity of the position function,. Use a graph to support your answer. Simply looking at the position graph, what do you think might be the significance of d (t)? Why? Answer: Acceleration is the rate at which is changing. Thus, a(t) = =. If velocity is measured in m/s, then acceleration would be. The of v(t) indicate possible changes in the sign of. Example: Find where the acceleration is changing if the velocity of an object is given by: v(t) = 4t 2. Groups, Part 1: Assigned function - d(t): v(t): a(t): Is the position graph concave up, concave down, or neither?

6 For t > 0, is velocity always positive or negative? Is the particle s speed increasing or decreasing? For t > 0, is acceleration positive, negative, or zero? Groups, Part 2: Position Is d(t) concave up, Velocity sign Object s speed Acceleration Function down, or neither? increasing? sign 8t

7 Questions to Consider: If velocity is constant, then acceleration is. What s the relationship between the concavity of d(t) and acceleration? Based on your observations of the sign value of velocity and acceleration, make a conjecture about what s necessary for an object s speed to be increasing: Independent Practice: The position function of an object is given by. a. In what intervals is the velocity decreasing? b. In what intervals is acceleration negative? c. Using parts (a) and (b), in what intervals is the object s speed increasing?

8 Group 1: Group 2: Group 3: Group 4: Group 5: 8t