02_Kinematics_sy14-15.pdf

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Chapter 2: Describing Motion – Kinematics in One Dimension

Kinematics

Name:

Contents:

Chapter Notes 2, 6

Practice Problems 4, 8-9

Understanding Concepts Practice 10-15

Ranking Exercises 16-19

Rocket Example 20

The Physics 500 Lab 22-24

Uniform Motion Lab 26-32

Uniform Accelerated Motion Lab 34-40

Reflex Activity 42

Graph Matching Activity 44-46

Picket Fence Lab 48-52

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Chapter 2: Describing Motion – Kinematics in One Dimension

Chapter Notes

Describing Motion – Kinematics in One Dimension

The Kinematic Equations

The kinematic equations are a set of equations, which can be utilized to determine unknown

information about an object’s motion if other information is known. The equations can be utilized for any motion, which can be described as being either a constant motion (an acceleration of 0 m/s2) or a constant accelerated motion. They can never be used over any time period during which the

acceleration is changing. Each of the kinematic equations includes four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object’s motion if other information is known.

𝒅 = 𝒗𝒊𝒕 + 𝟏 𝟐𝒂𝒕

𝟐

𝒗_𝒇𝟐 = 𝒗_𝒊𝟐+ 𝟐𝒂𝒅

𝒅 =𝒗𝒇+ 𝒗𝒊 𝟐 𝒕 𝒗𝒇= 𝒗𝒊+ 𝒂𝒕

Each symbol has its own specific meaning.

The symbol “d” stands for the displacement of the object. The symbol “t” stands for the time for which the object moved. The symbol “a” stands for the acceleration of the object.

The symbol “v” stands for the velocity of the object; a subscript of “i” after the v (as in vi)

indicates that the velocity value is the initial velocity value and a subscript of “f” (as in vf)

indicates that the velocity is the final velocity value.

Kinematics problem-solving strategy*

*Adapted from Randall D. Knight’s work in “Five Easy Lessons” 2004

Model: It is impossible to treat every detail of a situation. Simplify the situation with a model that captures the essential features. For example, the object in a mechanics problem is usually represented as a particle.

Visualize: This is the step where expert problem solvers put most of their effort.

 Draw a motion diagram and determine the acceleration vector at different point in the motion. This helps you visualize important aspects of the physics such as velocity and acceleration.  Draw a pictorial representation to establish known information, a coordinate system, and

symbols for later use. This helps you assess the information you are given and starts the process of translating the problem into symbols.

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Chapter 2: Describing Motion – Kinematics in One Dimension

Solve: Only after modeling and visualizing are complete is it time to develop a mathematical representation with specific equations that must be solved. The equations should use only symbols defined in the pictorial representation.

Assess: Is your result believable? Does it have proper units? Does it make sense?

Example

An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until it finally lifts off the ground. Determine the distance traveled before takeoff.

Motion Diagrams

1. Show the object’s position at equally space intervals of time. Represent the object as a dot (a particle). Five or six dots is usually sufficient to make the motion clear.

2. Draw velocity vectors. Velocity vectors connect each dot to the next.

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Chapter 2: Describing Motion – Kinematics in One Dimension Pictorial Diagrams

1. Sketch the situation. Not just any sketch. Show the object as the beginning of the motion, at the end, and at any point where the character of the motion changes. Very simple drawings are adequate.

2. Establish a coordinate system. Select your axes and origin to match the motion. 3. Define symbols. Use the sketch to define symbols representing quantities such as

position, velocity, acceleration, and time. Every variable used later in the mathematical solution should be defined on the sketch. Some will have known values, others are initially unknown, but all should be given symbolic names.

4. List the known information. Make a table of the quantities whose values you can determine from the problem statement or that can be found quickly with simple geometry or unit conversions. Some quantities are implied by the problem, rather than explicitly given. Others are determined by your choice of coordinate system.

5. Identify the desired unknowns. What quantity or quantities will allow you to answer the questions? These should have been defined as symbols in step 3. Don’t list every unknown; only the one or two needed to answer the question.

Mathematical Representation

1. Identify and list the equation that will be sued to determine unknown information from known information. The questions need to be answered in the affirmative for it to be the correct equation:

a. Does the equation contain my unknown? b. Do I know everything else in the equation?

2. Use appropriate algebraic steps to solve the equation for the unknown prior to substituting in known values

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Chapter 2: Describing Motion – Kinematics in One Dimension Practice Problems #1

1. How long does it take light traveling from the sun at 3.0 x 108 m/s to reach Jupiter, which is about 780 x 106 km away?

2. A car traveling 6.0 m/s is uniformly accelerating at a rate of 3.0 m/s2 for 15 seconds. What is its final velocity?

3. A ball rolling at 4.0 m/s accelerates uniformly down a hill at 5.5 m/s2 for 6.0 seconds. What is the final velocity?

4. How many seconds would it take a boat to accelerate from 13 m/s to 26 m/s over a distance of 1.25 km?

5. A racing car traveling initially at 8.0 m/s accelerates uniformly at 10.0 m/s2 for 5 seconds. How far does it travel in this time interval?

6. How fast does a car have to accelerate to go from 10.0 m/s to 30.0 m/s in 125 m?

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Chapter 2: Describing Motion – Kinematics in One Dimension Free Fall

A free falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects:

 Free-falling objects do not encounter air resistance.

 All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s for back-of-the-envelope calculations)

That is to say that any object, which is moving, and being acted upon only be the force of gravity is said to be “in a state of free fall.” Such an object will experience a

downward acceleration of -9.8 m/s2 (It is (-) because the direction is down). Whether the object is falling downward or rising upward towards its peak, if it is under the sole influence of gravity, its acceleration value is -9.8 m/s2. We use the symbol “g” to represent the acceleration of a freely falling object.

Representing Free Fall by Position-Time Graphs

Observe that the line on the graph curves. A curved line on a position versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s2), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object, the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward)

velocity.

Representing Free Fall by Velocity-Time Graphs

Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s2, downward), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object that is moving in the negative direction and speeding up is said to have a negative

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Chapter 2: Describing Motion – Kinematics in One Dimension Practice Problems #2

1. A baseball is hit straight up in the air with an initial velocity of 38 m/s. a. How long does it stay in the air?

b. How high does it go?

2. A racing car travels 480 km in 2.0 hours. Calculate the car’s average speed in km/h and m/s.

3. A car’s speed increases uniformly form 10 m/s, east, to 30 m/s, east in 5 seconds. a. Calculate the car’s acceleration.

b. How far does the car travel during the 5 second period?

4. A cart, initially at rest, accelerates at a rate of +3.0 m/s2 for 8.0s. a. How fast is the cart traveling after 8.0 s?

b. How far has the cart traveled?

5. A gun is fired and the bullet is accelerated in the gun barrel, which is 1.00 meter long. The bullet leaves the barrel with a velocity of 600.0 m/s. Calculate the acceleration of the bullet while it is in the barrel.

6. An electron in a vacuum tube is accelerated by a charged plate. The acceleration is

-7.0 x 108 m/s2 and takes place during an interval of 6.0 x 10-3 s. The speed of the electron after the acceleration takes place is 2.0 x 105 m/s. Calculate the speed of the electron before the acceleration.

7. A ball is dropped from a roof and takes 3.0 s to reach the ground. Calculate the elevation of the roof above the ground in meters [Neglect air resistance].

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Chapter 2: Describing Motion – Kinematics in One Dimension Practice Problems #3

1. An adult kangaroo can jump as high as 1.80 meters. With what initial velocity must a kangaroo leave the ground to reach this maximum height?

2. A bullet is shot straight up with a velocity of 58.8 m/s. a. Calculate the velocity of the bullet 1.5 s after firing.

b. How high above the gun is the bullet 1.5 s after firing?

c. How high is the bullet 9.0 s after firing?

d. Calculate the velocity of the bullet 9.0 s after firing.

3. A balloon is rising at 29.4 m/s. A stone falls from the balloon and reaches the ground in 20.0 s. How high is the balloon when the stone is dropped?

4. A boy standing on a 19.6 meter tall bridge sees a motorboat approaching the bridge at a

constant speed. When the boat is 27 meters from the bridge, the boy drops a stone to the water below. If the stone strikes the water 3.0 meters in front of the bow of the boat, at what speed is the boat traveling?

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Chapter 2: Describing Motion – Kinematics in One Dimension

Understanding Concepts

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Chapter 2: Describing Motion – Kinematics in One Dimension

Interpreting Graphical Data

Below is a velocity-time graph for a two-stage rocket. Answer the following questions based on

this graph.

1.

How fast was the rocket traveling when the second stage started?

2.

What was the fastest speed the rocket achieved? When did this occur?

3.

At what time did the rocket reach its highest altitude?

4.

What was the maximum altitude of the rocket?

5.

What was the acceleration of the capsule during the interval t = 100 sec to t = 300 sec?

6.

What was happening during the period t = 300 sec to t = 400 sec?

7.

What was the terminal velocity of the capsule just before “splashdown?”

8.

Write a short narrative describing the motion of the rocket during its entire journey.

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Chapter 2: Describing Motion – Kinematics in One Dimension

The Physics 500 Lab

Purpose

To compute the average speed of at least three different races and to participate in at least one race.

Required Equipment meterstick, stopwatch

Introduction

In this activity, you will need to think about what measurements are necessary to make in order to compute the average speed of an object. How does the average speed you compute compare with the maximum speed? How could you find the maximum speed of a runner or a car between stoplights?

Procedure

Step 1: Work in groups of about three students. Select instruments to measure distance and time. Develop a plan that will enable you to determine speed. Two students race each other in races such as hopping on one foot, rolling on the lawn, or walking backward. The third student collects and organizes data to determine the average speed of each racer. Repeat this process until each member of your group has a chance to be the timer. For the race in which you are the timer, record your plan and the type of race. Each member of the group will have different data.

When measurements are to be made in an experiment, a good experimenter organizes a table showing all data, not just the data that “seem to be right.” Record your data in Data Table A. Show the units you used as well as the quantities. For each measurement, record as many digits as you can read directly from the measuring instrument, plus one estimated digit. Then calculate the average speed for each student.

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Chapter 2: Describing Motion – Kinematics in One Dimension Data Table A:

Activity

Distance

Time

Speed

Analysis

1. How does average speed relate to the distance covered and the time taken for travel?

2. Should the recorded average speed represent the maximum speed for each event? Explain.

3. Which event had the greatest average speed in the class in miles per hour? (1.00 m/s = 2.24 mi/h)

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Chapter 2: Describing Motion – Kinematics in One Dimension

Name: _____________________________ My lab partner(s): ___________________

Date: _______________________ ___________________

Lab Title: ________________________________ ___________________

A Scientific explanation has four parts:



Claim: a conclusion to a question or problem; a statement that answers the question



Evidence: scientific data that supports the claim; data needs to be appropriate and

sufficient



Reasoning: a justification that links the evidence to the claim using scientific principles;

each piece of evidence may have a different justification for why it supports the claim



Rebuttal: describes alternative explanations, and provides counter evidence and

reasoning for why the alternative explanation is not appropriate

Evidence

Claim

Reasoning

Rebuttal

Claim, Evidence, Reasoning, Rebuttal

What do your calculations

mean?

What was the point of the lab?

Connect your evidence with your claim and

explain why.

Explain possible alternatives with

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Chapter 2: Describing Motion – Kinematics in One Dimension

Uniform Motion Lab

Introduction

In this lab you will observe and measure the motion of a motorized cart by marking its

position along a strip of tape at regular time intervals. Once you have a record of the cart’s

position, you can create graphs showing the cart’s motion graphically. In this lab you will create

graphs (displacement-time, velocity-time, and acceleration-time). Each graph will look different

but each will represent the same motion in its own way. From each graph we will learn how to

determine the cart’s displacement, velocity, and acceleration by reading data from the y-axis,

looking at the line of best fit’s shape, determining the line of best fit’s slope, and determining

the area under the curve. By doing this, you will learn the nature of UNIFORM MOTION.

UNIFORM MOTION means constant motion. An object experiencing UNIFORM MOTION travels

equal displacement in equal intervals of time, at a constant velocity, and with zero acceleration.

Objective

In this lab, you will:

1.

Record the positions of two cars of different speed, which are undergoing uniform

motion.

2.

Create displacement, velocity, and acceleration graphs showing uniform motion.

3.

Learn what the shapes, slopes areas, and y-values of each graph represents.

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Chapter 2: Describing Motion – Kinematics in One Dimension

Procedure

1.

Within your group, appoint a Car Operator, a Timer, and a Marker.

2.

Tape a strip of ticker tape along the entire length of your lab.

3.

Mark a starting point on the ticker tape approximately 20 cm from one end of the

table. Label this line “Starting Line.” Also label this side of the ticker tape “Slow Car.”

4.

Turn the “Slow Car” on and place it on the table behind the starting line.

5.

When the front of the car crosses the starting line, the Timer starts the stopwatch.

6.

Every second, the Timer shouts “Mark!” to the Marker.

7.

The Marker makes a mark on the ticker tape at the position of the front of the car.

8.

Make as many marks as you can before the car makes it to the end of the ticker

tape. Stop the car before it falls off the table, please.

9.

Turn off the car.

10.

Repeat steps 2 through 9 using the “Fast Car.” Label this ticker tape side “Fast Car.”

11.

Use a meter stick or ruler to measure the distance between marks. Record your

measurements in the tables below. You may not necessarily fill the entire table. It

depends on how many marks you attained.

12.

Complete both tables by:

a.

Calculating the car’s “Displacement” from the starting line by adding the next

“distance between marks” value to the previous “Displacement” value.

b.

Calculating the car’s “Velocity” by taking the “distance between the marks”

and dividing it by 1 second. [Yes, this is as easy as it sounds.]

c.

Calculating the car’s “Change in Velocity” by taking the difference between

the two “velocity” measurements.

d.

Calculating the car’s “Acceleration” by taking the “change in velocity”

calculation and dividing it by 1 second.

13.

Make a displacement-time graph showing both cars’ motion. Use different colored

pencils for each data set and include a legend. Record the Time as the independent

variable and the Displacement as the dependent variable. Draw a line of best fit.

14.

Make a velocity-time graph showing both cars’ motion. Use different colored pencils

for each data set and include a legend. Record the Time as the independent variable

and the Velocity as the dependent variable. Draw a line of best fit.

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Chapter 2: Describing Motion – Kinematics in One Dimension

SLOW CAR DATA TABLE

Mark

Time (s)

Distance

between

marks (cm)

Displacement

(cm)

Velocity

(cm/s)

Change in

Velocity

(cm/s)

Acceleration

(cm/s

)

Start

0

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0.00

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1

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Chapter 2: Describing Motion – Kinematics in One Dimension

FAST CAR DATA TABLE

Mark

Time (s)

Distance

between

marks (cm)

Displacement

(cm)

Velocity

(cm/s)

Change in

Velocity

(cm/s)

Acceleration

(cm/s

)

Start

0

---

0.00

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---

1

1

2

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Chapter 2: Describing Motion – Kinematics in One Dimension

Guiding Questions (to help with your lab report)

1.

Look at the Displacement-time graph. What is the shape of the displacement-time graph

for an object undergoing uniform motion?

a.

Does the graph have a constant slope? ____________ If so, the cart is

undergoing uniform motion.

b.

Calculate the slope of the line. Be sure to include units in your calculation.

c.

What physical quantity does the slope of the displacement-time graph

represent?

d.

What does the graph’s slope tell you about the car’s velocity? How would you

recognize constant velocity on a displacement-time graph?

2.

Look at the Velocity-time graph. What is the shape of the velocity-time graph for an

object undergoing uniform motion?

a.

Calculate the slope of the line. Be sure to include units in your calculation.

b.

What physical quantity does the slope of a velocity-time graph represent?

c.

What does the graph’s slope tell you about the car’s acceleration?

d.

Calculate the area under the curve for the first 5 seconds of motion. Be sure to

include units in your calculation.

e.

What physical quantity does the area under the curve of a velocity-time graph

represent?

f.

Compare your value for the area under the curve (step 2d) with the car’s

displacement at t = 5 s using your displacement-time graph. How do they

compare?

g.

How would you recognize constant velocity on a velocity-time graph?

3.

Look at the acceleration-time graph. What is the shape of the acceleration-time graph

for an object undergoing uniform motion?

a.

Calculate the area under the curve for the first 5 seconds (if there is any). Be sure

to include units in your calculation.

b.

What physical quantity does the area under the curve of an acceleration-time

graph represent?

c.

Compare your value for the area under the curve (step 3a) with the car’s velocity

at t = 5 s, using your velocity-time graph. How do they compare?

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Chapter 2: Describing Motion – Kinematics in One Dimension

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Chapter 2: Describing Motion – Kinematics in One Dimension

Name: _____________________________ My lab partner(s): ___________________

Date: _______________________ ___________________

Lab Title: ________________________________ ___________________

A Scientific explanation has four parts:



Claim: a conclusion to a question or problem; a statement that answers the question



Evidence: scientific data that supports the claim; data needs to be appropriate and

sufficient



Reasoning: a justification that links the evidence to the claim using scientific principles;

each piece of evidence may have a different justification for why it supports the claim



Rebuttal: describes alternative explanations, and provides counter evidence and

reasoning for why the alternative explanation is not appropriate

Evidence

Claim

Reasoning

Rebuttal

Claim, Evidence, Reasoning, Rebuttal

What do your calculations

mean?

What was the point of the lab?

Connect your evidence with your claim and

explain why.

Explain possible alternatives with

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Chapter 2: Describing Motion – Kinematics in One Dimension

Uniform Accelerated Motion Lab

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Chapter 2: Describing Motion – Kinematics in One Dimension

Name: _____________________________ My lab partner(s): ___________________

Date: _______________________ ___________________

Lab Title: ________________________________ ___________________

A Scientific explanation has four parts:



Claim: a conclusion to a question or problem; a statement that answers the question



Evidence: scientific data that supports the claim; data needs to be appropriate and

sufficient



Reasoning: a justification that links the evidence to the claim using scientific principles;

each piece of evidence may have a different justification for why it supports the claim



Rebuttal: describes alternative explanations, and provides counter evidence and

reasoning for why the alternative explanation is not appropriate

Evidence

Claim

Reasoning

Rebuttal

Claim, Evidence, Reasoning, Rebuttal

Connect your evidence with your claim and

explain why.

Explain possible alternatives with

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Chapter 2: Describing Motion – Kinematics in One Dimension

Reflex Activity

Date: _____________________ Pd: ___________

Objectives:

1. To see how fast you can pinch your fingers closed when a ruler is dropped between them. 2. To measure your reflexes.

Procedure:

1. Have your partner place a ruler with the 0.0 cm reading between your thumb and index finger. Set your thumb about 3.0 cm apart from your index finger.

2. When your partner, without indication, drops the ruler quickly grasp the ruler. 3. Record the reading to the nearest centimeter.

4. Repeat steps 1 through 3 two more times.

Data:

Distance ruler dropped (trial 1) d1 =

Distance ruler dropped (trial 2) d2 =

Distance ruler dropped (trial 3) d3 =

Initial velocity of ruler vi =

Acceleration of ruler a = Unknown variable

Calculations:

1. By looking at the data section, pick an equation that will allow you to solve for the unknown. Then solve for the unknown, leaving the expression in variable form.

2. For each trial, solve for the unknown variable. Show your work. a. Trial 1:

b. Trial 2:

c. Trial 3:

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Chapter 2: Describing Motion – Kinematics in One Dimension

Graph Matching Lab

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Chapter 2: Describing Motion – Kinematics in One Dimension

Name: _____________________________ My lab partner(s): ___________________

Date: _______________________ ___________________

Lab Title: ________________________________ ___________________

A Scientific explanation has four parts:



Claim: a conclusion to a question or problem; a statement that answers the question



Evidence: scientific data that supports the claim; data needs to be appropriate and

sufficient



Reasoning: a justification that links the evidence to the claim using scientific principles;

each piece of evidence may have a different justification for why it supports the claim



Rebuttal: describes alternative explanations, and provides counter evidence and

reasoning for why the alternative explanation is not appropriate

Evidence

Claim

Reasoning

Rebuttal

Claim, Evidence, Reasoning, Rebuttal

What do your calculations

mean?

What was the point of the lab?

Connect your evidence with your claim and

explain why.

Explain possible alternatives with

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