01_u3 teacher notes

28
Unit 3: Uniformly Accelerated Particle Model Teacher Notes Uniformly Accelerated Particle Model Teacher Notes INSTRUCTIONAL GOALS 1. Concepts of acceleration, average vs. instantaneous velocity Contrast graphs of objects undergoing constant velocity and constant acceleration Define instantaneous velocity (slope of tangent to curve in vs. t graph) Distinguish between instantaneous and average velocity Define acceleration, including its vector nature Motion map now includes acceleration vectors 2. Multiple representations (graphical, algebraic, diagrammatic) Introduce stack of kinematic curves position vs. time (slope of tangent = instantaneous velocity) velocity vs. time (slope = acceleration, area under curve = change in position) acceleration vs. time (area under curve = change in velocity) Relate various expressions 3. Uniformly Accelerating Particle model Kinematical properties of objects undergoing uniformly accelerated motion Derive the following relationships from vs. t and vs. t graphs definition of average acceleration linear equation for a vs. t graph generalized equation for any t i to t f interval parabolic equation for an vs. t graph generalized equation for any t i to t f interval time independent equation ©Notre Dame Modeling Project 2012 1ICP RAP Uniform Acceleration - Teacher Notes

Upload: bobkane

Post on 26-Dec-2015

321 views

Category:

Documents


9 download

DESCRIPTION

Notes on physics section 1 unit 3

TRANSCRIPT

Page 1: 01_U3 Teacher Notes

Unit 3: Uniformly Accelerated Particle Model Teacher Notes

Uniformly Accelerated Particle ModelTeacher Notes

INSTRUCTIONAL GOALS1. Concepts of acceleration, average vs. instantaneous velocity

Contrast graphs of objects undergoing constant velocity and constant accelerationDefine instantaneous velocity (slope of tangent to curve in vs. t graph)Distinguish between instantaneous and average velocityDefine acceleration, including its vector natureMotion map now includes acceleration vectors

2. Multiple representations (graphical, algebraic, diagrammatic)Introduce stack of kinematic curves

position vs. time (slope of tangent = instantaneous velocity)velocity vs. time (slope = acceleration, area under curve = change in position)acceleration vs. time (area under curve = change in velocity)

Relate various expressions

3. Uniformly Accelerating Particle modelKinematical properties of objects undergoing uniformly accelerated motionDerive the following relationships from vs. t and vs. t graphs

definition of average acceleration

linear equation for a vs. t graphgeneralized equation for any ti to tf interval

parabolic equation for an vs. t graph

generalized equation for any ti to tf interval

time independent equation

4. Analysis of free fall

5. SoftwareConceptual Kinematics Tutorial Graphs and Tracks (Physics Academic Software)Equivalent online versions are available

©Notre Dame Modeling Project 2012 1 ICP RAP Uniform Acceleration - Teacher Notes

Page 2: 01_U3 Teacher Notes

SEQUENCE

1. Wheel Lab: Motion of an object on an incline; instantaneous velocity; slopes of tangents; Tangent Practice

2. Worksheet 1a-f: Uniformly Accelerated Motion Worksheet. Relate the average velocity for a time interval (

) to the instantaneous velocity ( ) at the middle of that time interval.

3. Worksheet 2a: Accelerated Motion Representations

4. Worksheet 2b: Stacks of Kinematic Curves

5. Quiz 1: Stacks of x-t, v-t, and a-t graphs

6. Deployment Labs: Freefall and top of the flight; Fan cart motion

7. Worksheet 3: Quantitative Acceleration Problems

8. Quiz 2: Velocity vs. Time Graphs

9. Free fall of Wile E Coyote Planet Newtonia

10. Free fall of Wile E Coyote Newtonia’s Moon

11. Review Sheet

12. Uniformly Accelerated Particle Model Test

©Notre Dame Modeling Project 2012 2 ICP RAP Uniform Acceleration - Teacher Notes

Page 3: 01_U3 Teacher Notes

1. Lab: Wheel LabDevelop the “position proportional to time-squared” relation during whiteboarding.Post-lab: Use tangents to the x-t graph curve to create a v-t graph and define instantaneous velocity.

Lab Notes: Wheel Lab

Apparatus

©Notre Dame Modeling Project 2012 3 ICP RAP Uniform Acceleration - Teacher Notes

Page 4: 01_U3 Teacher Notes

Wheel (build from 4inch hole saw cut-out, dowel, golf tees)Track (two lengths of electrical conduit)Masking tapeTicker tapeMetronome

Pre-lab discussion

Let a wheel roll down an inclined rail and ask students for observations. Record all observations. To proceed, they must mention something to the effect that the ball speeds up as it rolls down.

To obtain a finer description, ask students which observations are measurable. Make sure they include the observation that the ball speeds up as it rolls down the rail. (Do not let them state the ball accelerates since we haven't defined acceleration yet!)

Ask them how they can measure speed directly. Lead them to the conclusion that they cannot, but that they can measure position and time. Note: Since this lab is so similar to the BB lab and other motion labs, this should be a matter of review to most of them.

Students should mark the position of the object at equal time intervals. Time should be plotted as the independent variable. One key difference between this lab and the past motion labs is that the wheel must start from

position 0 with a velocity of 0. Students don’t need to know why yet. They should be told to make a mark on the tape near the top of the track to call position 0. Upon hearing a beat of the metronome, they will release the wheel from rest. Marks for position should be made on subsequent beats.

Speeding Up Wheel Lab

Purpose: to develop graphical, mathematical, diagrammatical, and verbal representations for an object that starts from rest and gradually increases in speed.

Procedure:

Use a marker on the paper tape to mark the position of the axle at equal time intervals.IMPORTANT: Your first dot must be for time = 0, position = 0, and velocity = 0.

Data Tables:

Low incline (one brick) High incline (two bricks)Height of raised end = ______ Height of raised end = ______

©Notre Dame Modeling Project 2012 4 ICP RAP Uniform Acceleration - Teacher Notes

Time (s) Position (m)

Wheel and axle

brickspipes

paper strip on board beside pipes

Line on wheel must point down the incline.

Page 5: 01_U3 Teacher Notes

Graphs:Create a position-time graph for each set of data individually. Do any necessary test plots. Print. Save your graphs to the computer. Next, print each position-time graph individually so that it is the only thing on the page, in landscape orientation. (Use File -> “Print Graph”)

Lab performance notes

If wheels and axles aren’t available, other setups can be used such as a cart rolling down a track, a bowling ball rolling down an access ramp, or a disc and axle rolling down a ramp of two parallel pieces of conduit pipe. Just make sure the object moves slowly enough for students to accurately mark position and time.

Timing variations could include using water clocks, pendulums and metronomes in addition to stopwatches. Calling the unit of time something like “beats” instead of “ticks” reduces confusion between units and variables when doing mathematical analysis.

Two angles of inclination should be used. Position time data can be plotted on one set of axes. But instruct students to make the graph a large as possible and to orient the graph so that the long side of the paper is the time axis.

©Notre Dame Modeling Project 2012 5 ICP RAP Uniform Acceleration - Teacher Notes

Page 6: 01_U3 Teacher Notes

Sample Data: Wheel LabWheel Lab Data Tables with columns for kinematics graph

Wheel Lab Graphs of Position vs. Time, Velocity vs. Time and Position vs. Time2

©Notre Dame Modeling Project 2012 6 ICP RAP Uniform Acceleration - Teacher Notes

Wheel/Axle on Low Inclinetime position ∆time ∆position avg velocity midtime time^2

(beats) (cm) (beats) (cm) (cm/beat) (beats) (beats^2)0.0 0.0 0.01.0 1.0 1.0 1.0 1.0 0.5 1.02.0 3.2 1.0 2.2 2.2 1.5 4.03.0 5.6 1.0 2.4 2.4 2.5 9.04.0 9.6 1.0 4.0 4.0 3.5 16.05.0 13.5 1.0 3.9 3.9 4.5 25.06.0 18.3 1.0 4.8 4.8 5.5 36.07.0 24.9 1.0 6.6 6.6 6.5 49.08.0 31.0 1.0 6.1 6.1 7.5 64.09.0 38.1 1.0 7.1 7.1 8.5 81.0

10.0 46.2 1.0 8.1 8.1 9.5 100.011.0 55.4 1.0 9.2 9.2 10.5 121.012.0 65.0 1.0 9.6 9.6 11.5 144.013.0 76.0 1.0 11.0 11.0 12.5 169.014.0 86.8 1.0 10.8 10.8 13.5 196.015.0 99.2 1.0 12.4 12.4 14.5 225.0

Page 7: 01_U3 Teacher Notes

©Notre Dame Modeling Project 2012 7 ICP RAP Uniform Acceleration - Teacher Notes

Page 8: 01_U3 Teacher Notes

Post-lab discussion

From the lab, the students have the following graph:

Focus the whiteboard discussion on their experimental procedure and the verbal interpretation of the parabolic x-t graph. Students should be able to describe that the displacement during each time interval increases over the previous time interval. Since the object travels greater distances in each successive time interval, the velocity is increasing.

Post-lab extensionDiscussionContrast the x vs t graph for this lab with the one obtained in unit 1.

One can speak of the average velocity as the slope of the graph (above left) because the slope of a straight line is constant. It doesn't matter which two points are used to determine the slope.

On the other hand, one could speak of the average velocity of the object in the graph to the right, but since the object started very slowly and steadily increased its speed, the term average velocity has little meaning.

©Notre Dame Modeling Project 2012 8 ICP RAP Uniform Acceleration - Teacher Notes

x(cm)

t t

Unit 1 Unit 2

t

x

t

Page 9: 01_U3 Teacher Notes

What would be more useful is to have a way of describing the object's speed at a given instant (or as Arons terms it: clock reading). To develop this idea, you must show that, as you shrink the timeinterval t over which you calculate the average velocity, the secant (line intersecting the curve at two points) more closely resembles the curve during that interval.

That is, the slope of the secant gives the average velocity for that interval. As the interval gets shorter and shorter, the secant more closely approximates the curve. Thus, the average velocity of this interval becomes a more and more reasonable estimate of how fast the object is moving at any instant during this interval.

As one shrinks the interval, t to zero, the secant becomes a tangent; the slope of the tangent is the average velocity at this instant, or simply the instantaneous velocity at that clock reading.

Student activityUsing the position vs. time graphs the students produced in the lab, students should construct at least five tangents to the curve and determine the slope of each tangent. This is the reason students made graphs as large as possible with time on the long side of the paper. The students should then make a new graph of instantaneous velocity vs time. A plot of instantaneous velocity ( , instead of ) vs time should yield a straight line. The slope of this

line is . That is, the change in velocity during a given time interval is defined to be the average

acceleration. Students need to understand the units for acceleration. In this lab, the units for the slope of the graph will be . Students must be able to state that a slope of 5cm/b/b for example, means that the wheel’s velocity changed 5cm/b for each beat of time. Many times, cm/b/b is written as cm/b2. It will reduce confusion if the acceleration is stated as “centimeters per beat per beat” even if it is written as cm/b2.

The graph of vs t for the wheel lab, shown to the right, is linear meaning that acceleration is constant. If the acceleration is constant, then the average acceleration for the entire time interval is equal to the instantaneous acceleration at any given clock reading. The equation for the line can be written as , where is the y-intercept. It is important to define the acceleration this way, and then show examples of vs t graphs in which the acceleration is negative. Acceleration will be negative if the quantity is negative.

©Notre Dame Modeling Project 2012 9 ICP RAP Uniform Acceleration - Teacher Notes

x

t

x

t

x

x

t

x

t

x

tt

x

t

t

Page 10: 01_U3 Teacher Notes

On the vs t graphs shown to the right, tangent lines were drawn to show the instantaneous velocity at two different clock readings. In both cases, is negative, yet very different situations are being represented. In the graph on the left, an object slows down while moving in the positive direction. In the graph on the right, the object speeds up while moving in the negative direction. In both graphs, the acceleration is in the negative direction. We advise against the use of the term deceleration, because students invariably think that this term implies negative acceleration means slowing down; the two conditions are not synonymous.

Generalizing the linear equation from the velocity vs. time graph for any time interval ti to tf yields . The development of this expression is provided below to clarify the use of t as opposed to t.

Lab SummaryAt this point, the students have six graphs and multiple equations on multiple sheets of paper. A lab summary sheet is provided in the materials so that students can transfer all of their critical conclusions onto a single sheet of paper.

Teacher background

©Notre Dame Modeling Project 2012 10 ICP RAP Uniform Acceleration - Teacher Notes

tt

x

v1

v1

v2

v2

x

Page 11: 01_U3 Teacher Notes

The slope is defined to be average velocity.

Eq.1

Equation of the lineEq.2

Generalize the equation for the interval ti to tf.At tf:

Eq.3At ti:

Eq.4Subtract equation 4 from 3:

Eq.5

The slope is defined to be average acceleration.

Eq.6

Equation of the lineEq.7

Generalize the equation for the interval ti to tf.At tf:

Eq.8At ti:

Eq.9Subtract equation 9 from 8:

Eq.10

Post-lab Extension (Development of Kinematic Expressions)Developing the remaining kinematic equations involves finding the area under a v-t graph and algebraic combination of equations. Depending on the ability of your students, various levels of guidance can be provided to help the students derive the equations themselves. Don’t get hung up on the algebra. Focus on the physics. The displacement of a uniformly accelerating object is equivalent to the area under the v-t graph. In this situation, we are interested in the displacement during the time interval ti to tf.

Page 12: 01_U3 Teacher Notes

Area of region A:1/2 height x base area of a triangle

substitute

Area of region Blength x width area of a rectangleThe velocity at the horizontal axis is zero;

=

Page 13: 01_U3 Teacher Notes

The total displacement is equal to A + B.

Rearranging:Eq.11

Combining equations 6 and 11 produces a time-independent kinematics expression.

Eq. 6

Rearrange:

; Eq. 12

Eq. 11

Substitute equation 12 into equation 11:

Multiply both sides by

Multiply out the terms on the right.

Simplify the right side of equation

Rearrange: Eq. 13

Summary of mathematical models:

Eq. 6 definition of average acceleration

Eq. 7 linear equation for a v-t graphEq. 10 generalized equation for any ti to tf interval

parabolic equation for an x-t graph

Eq. 11 generalized equation for any ti to tf interval

Eq. 13 algebraic combination of equations 3 and 5

Optional Post Lab Extension: Linearizing the vs. Graph

Before embarking on the algebraic journey to derive Eq. 11 and Eq. 13 above, you may want to do further analysis of the curved position vs. time graph. In the uniform motion experiments, it became clear that if the starting position of the object was zero and the object moved at a constant velocity, then the position of the object was directly proportional to the time. If twice as much time elapsed, the object’s position was twice as far from zero.

©Modeling Instruction 2010 13 Unit III Teacher Notes v3.0

Page 14: 01_U3 Teacher Notes

Upon completing the graph of position vs. time for the wheel lab, students should see that the relationship between the two variables is different. At some clock reading of t, the wheel is at positon x. At a clock reading of 2t, will the wheel be at 2x? How about at a clock reading of 3t?

Some students can recognize that the shape of the x-t graph is parabolic. A few of those students might even remember the general form of a quadratic equation (y=Ax2+Bx+C). But probably no one will be able to find the values of the coefficients of a quadratic equation given the wheel lab x-t graph.

Student ActivityHave students add a third column to their wheel lab data tables—a column for t2. Ask students in what units t2 should be measured. Fill in the column for t2 by squaring the clock readings in the t column. Students can now make a graph of position vs. time2. The graph should look like the one to the right.

©Modeling Instruction 2010 14 Unit III Teacher Notes v3.0

t2

Page 15: 01_U3 Teacher Notes

DiscussionFrom the graph, it is clear that position is directly proportional to the square of the time. In other words, if at some clock reading of t, the wheel is at position x, than at 2t, the object will be at 4x. At 3t, the object will be at 9x. Upon performing a mathematical analysis of the graph of vs. students will obtain the following mathematical model: where k is the slope of the graph. Students should be asked about the units of the slope and what those units mean. A few students will notice that the units of the graph are cm/b2. Ask what physical quantity has those units. Comparing the slope of this graph and the slope of the v-t graph from this unit should lead students to the understanding that the slope of the x-t2 graph is equal to 1/2 of the acceleration of the object. Thus, the general form of the equation for the x-t2 graph becomes . Make it clear, however, that this relationship is only true if the object starts from rest at position zero. When students ask what happens if the object starts somewhere else and is already moving at time zero, tell them you are glad that they asked and then embark on your algebraic journey.

2. Worksheet 1: Uniformly Accelerated Motion Worksheet Students will see that the creating velocity vs. time graphs by finding the slope of a sequence of tangents to the x-t graph is a bit tedious. However, the instantaneous velocity can also be found mathematically from a sequence of position-time data. The desired goal of the activity is to establish that for uniformly accelerated motion, the average velocity for a time interval is equal to the instantaneous velocity at the clock reading in the middle of the time interval. This idea is important since it is the same algorithm used by computer motion analysis programs such as Mac Motion, Logger Pro, and Science Workshop. Students can use the reasoning developed here to analyze the motion of a picket fence falling through a photogate.

Use a thought experiment with small numbers (an acceleration of 2 m/s2) to build the first two columns of simulated position vs. time “data,” leaving a space between each row. Utilize the expression to lead the students to identifying the instantaneous velocity at each time. Since the initial velocity is zero, the average velocity from t = 0 to t = tf is half the instantaneous velocity at t = tf and the displacement is the average velocity times t. For example, at t = 3 s the instantaneous velocity is 2 m/s2 x 3 s or 6 m/s. The average velocity from t = 0 to t = 3 s is 3 m/s and the displacement is 3 m/s x 3 s or 9 m.

t(s)

x(m)

t(s)

x(m)

vaverage

(m/s)tmiddle of interval

(s)0.0 0.0

1.0 1.0

2.0 4.0

3.0 9.0

4.0 16

5.0 25

6.0 36

©Modeling Instruction 2010 15 Unit III Teacher Notes v3.0

Page 16: 01_U3 Teacher Notes

After completing the first two columns of “data,” analyze the resulting x-t graph. The goal is to find the correlation between average velocity during a time interval and instantaneous velocity at a clock reading.

The average velocity from t = 2 s to t = 4 s:

From the graph, it appears that the slope of the chord connecting t = 2 s to t = 4 s has the same slope as the tangent to the curve at t = 3 s. The slope of the tangent at t = 3 s:

Let’s try the comparison of average and instantaneous velocities on a different part of the graph.

From the graph, it appears that the slope of the chord connecting t = 4 s to t = 6 s has the same slope as the tangent to the curve at t = 5 s. The slope of the tangent at t = 5 s:

Although the examples above do not constitute a rigorous proof, they provide a strong argument for the following conclusion: for uniformly accelerated motion, the average velocity for a time interval is equal to the instantaneous velocity at the clock reading in the middle of the time interval.

©Modeling Instruction 2010 16 Unit III Teacher Notes v3.0

Page 17: 01_U3 Teacher Notes

Calculations can now be made to fill in the rest of table below:

t(s)

x(m)

t(s)

x(m)

vaverage

(m/s)tmiddle of interval

(s)0.0 0.0

1.0 1.0 1 0.51.0 1.0

1.0 3.0 3 1.52.0 4.0

1.0 5.0 5 2.53.0 9.0

1.0 7.0 7 3.54.0 16

1.0 9.0 9 4.55.0 25

1.0 11.0 11 5.56.0 36

Using the idea that the average velocity during a time interval is equal to the instantaneous velocity at the clock reading in the middle of the time interval, an instantaneous velocity vs time graph can produced.

The students now have a mathematical way of producing a v-t graph from their measurements of position and time. Analyzing the v-t graph allows determination of acceleration and displacement with or without an initial velocity. With this background, students can begin the uniformly accelerated motion worksheet.

After completing and whiteboarding the worksheet, students should revisit the wheel lab. They now have the tools to do a much better graph of velocity vs. time than when they had to find slopes of tangent lines. They should be assigned to take their original position and time data for both inclines and make a velocity vs. time graph and a position vs. time2 graph. On the following pages are actual data obtained from a wheel and axle lab and graphs of x-t, x-t2 and v-t.

©Modeling Instruction 2010 17 Unit III Teacher Notes v3.0

Page 18: 01_U3 Teacher Notes

3. Worksheet 2 Graphs and Tracks

4. Stacks of Kinematic Curves

5. Quiz 1: Stacks of x-t, v-t, and a-t graphs

6. Deployment Labs: Freefall

7. Worksheet 3: Quantitative Acceleration Problems

8. Uniformly Accelerated Particle Model Quiz 2: Velocity vs. Time Graphs

9. Free fall on Planet Newtonia

10. Free fall on Newtonia’s Moon

11. Review Sheet

12. Model So Far…

1. The slope of a position-time graph is the velocity. If the position-time graph is curved, the slope of a line tangent to the curve tells you the velocity at that time. The velocity at a time is called instantaneous velocity. 2. In general, acceleration is the rate of change in velocity, which is the slope of a velocity-time graph.Mathematically, a = v/t 3. The velocity of a uniformly accelerating object increases or decreases by equal amounts each second. Therefore, the general linearized equation for a v vs. t graph is:  4. The area under a velocity-time graph is the object's change in position, or displacement.

©Modeling Instruction 2010 18 Unit III Teacher Notes v3.0

Page 19: 01_U3 Teacher Notes

Mathematically, x = ½ at2 + vit 5. The motion map for uniformly accelerated motion features dots whose successive spacing increases or decreases. Draw the dots for the location of the object at equally spaced time intervals, then add the velocity and acceleration vectors. 

 6. The acceleration due to gravity (g = 9.8 m/s2) near the surface of the earth is constant for all objects, ignoring effects of air resistance. 

 13. Uniformly Accelerated Particle Model Test

©Modeling Instruction 2010 19 Unit III Teacher Notes v3.0