chapter 1: concepts of motion - capca€¦ · 2 chapter 1: concepts of motion 1.1 motion diagrams...

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Chapter 1: concepts of motion

1.1 Motion Diagrams pp. 2-51.1 Motion Diagrams pp. 2-51.2 The Particle Model pp. 5-61.2 The Particle Model pp. 5-61.3 Position and Time pp. 6-101.3 Position and Time pp. 6-101.4 Velocity pp. 11-131.5 Linear Acceleration pp. 13-171.6 Motion in One Dimension pp. 17-201.7 Solving Problems in Physics pp. 20-241.8 Units and Significant Figures pp. 24-28

ATTENTION: The first number refers to the chapter; the second number refers tothe section. For example, 1.1 means section 1 of chapter 1 in your textbook.

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1.1 Motion Diagrams pp. 2-51.1 Motion Diagrams pp. 2-5

A motion diagram is a composite “photo”showing the object’s positions at severalequally spaced instants of time.

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A motion diagram of a moving object is simplya stack of photographs of the object, taken atdifferent times, and put above each other

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Time for a clicker question

CLICTION time

ATTENTION: Clictions will be numbered per chapter.For example, CLICTION 1.3 means clicker question number 3 of chapter 1.

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Which car is going faster, 1 or 2? Assume there are equalintervals of time between the frames of both movies.

Car 1 Car 2

Cliction 1.1

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Which car is going faster, 1 or 2? Assume there are equalintervals of time between the frames of both movies.

Car 1 Car 2

2 is going faster

Cliction 1.1

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1.2 The particle Model pp. 5-61.2 The particle Model pp. 5-6

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It is often a very good model to think of an objectas a structure-less point (particle) placed at its

center of mass (will be explained in more detail inlater chapters)

ATTENTION: this is not a rolling ball! It is adot representing a car coming to a stop as it

moves from the left to the right of the diagram.

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Three motion diagrams are shown. Which is a dust particlesettling to the floor at constant speed, which is a balldropped from the roof of a building, and which is adescending rocket slowing to make a soft landing onMars?

1. (a) is dust, (b) is ball, (c) is rocket

2. (a) is ball, (b) is dust, (c) is rocket

3. (a) is rocket, (b) is dust, (c) is ball

4. (a) is rocket, (b) is ball, (c) is dust

5. (a) is ball, (b) is rocket, (c) is dust

Cliction 1.2

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1.3 Position and Time pp. 6-101.3 Position and Time pp. 6-10

How can wetrack the object

In SPACE?

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The concepts of motion are based onfunctional relations:

position x of an object at a time t: x(t) velocity v of an object at a time t: v(t) acceleration a of an object at a time t: a(t)

In this chapter we will explore theirkinematical meaning

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For a car that remains on a straight road wewrite its position x at different times t1, t2 ..as x1, x2 ..

Generally, moving objects donot stay on a straight line butmay also fly up and down orleft/right

We then have to describe its position at a timet1 as a vector r1 = (x1, y1)Alternatively, r1 can be specified as a pair(distance from origin, direction from origin)


→→

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Both description have their advantages

r1 =(distance from origin, direction from origin)

is more intuitive. It directly tells us the lengthand direction of the vector, e.g., (100m, NW)

r1 = (x1, y1) is like a coordinate grid. This is very convenient for adding or subtractingvectors


→

→

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Example: Calgary's Avenue/Street system wherethe length is measured in “blocks”:


→

x

y r1 = (2 blks,5 blks)→

r2 = (-7 blks, -4 blks )

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Very often we will consider differences of vectorslike the distance travelled between two conse-cutive frames of the motion diagram


r1 = r0 + Δr→ → →

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In a motion diagram we can think of the positionvector rn+1 as being the sum of the previousdifference vectors:


→

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1.4 Velocity pp. 11-131.4 Velocity pp. 11-13

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The velocity of an object is a measure of howquickly an object can go from one position toanother during a time interval Δt.

Since its position is a vector, the velocity v of aparticle is a vector, too:


→

→

A particle moves from r1 to r2 in time Δt so that→ →

The SI unit of a velocity is m/s (meter per second)

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The velocity allows us to calculate the positionvector of a particle after it has been movingfor a time Δt:

Chapter 1: concepts of motionPh

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VERY IMPORTANTVERY IMPORTANT !! !!

Download (from course webpage) additionalDownload (from course webpage) additionalnotes on Vectors (week 1)notes on Vectors (week 1)

ClictionsClictions 1.3 and 1.4 are part of these additional1.3 and 1.4 are part of these additionalnotesnotes

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End of Week 1

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1.5 Linear Acceleration pp. 13-171.5 Linear Acceleration pp. 13-17

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Acceleration is one of the most importantconcepts in Physics.

It describes the change of the velocity from onetime step (frame in a motion diagram) to the next

Since the difference vector between two framesis proportional to the velocity,

Δr = v Δt ,

we can find the acceleration by comparingneighbouring difference vectors


→→

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Definition of the acceleration vector a:


→→

→

a can be visualized using a motion diagram→

Bombardier beetlewww.pnas.org/content/vol96/issue17/images/large/pq1792714001.jpeg

DESY particle accelerator, Hamburg

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1.6 Motion in one dimension p. 17-20


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→

→

Mathematics can be used in Physics becausethe systems (a ball, a wheel) can be definedvery accurately

Its purpose is different: Physicists are too lazyto memorize all different motion diagrams.Mathematics allows one to describe manydifferent types of motion with a single equation

Mathematics makes things easier , that's whyit is used in the first place.

?


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→

→

To describe motion, we have to learn how tocalculate with vectors.

Start with the simplest case: motion in one dimension.

Conveniently we can choose our coordinate system in such a way that the motion is along the x-axis.

Instead of a vector we then can consider only its component x (position along x axis),

vx (x-velocity), and ax (x-acceleration)

X-components


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→

→

Tactic box 1.4 (page 18 in your textbook):

This is VERY important (see next slide)

Y-axis for vertical motion!X-axis for horizontal motionand along an inclined plane!


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→

→

Be careful when interpreting the sign of theacceleration:

Object is speeding up ⇔ vx and ax have the same sign Object is slowing down ⇔ vx and ax have opposite sign constant speed ⇔ ax =0

It always helps to think of the acceleration as an underlying force!

Remember: F = m a


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→

→

Now that we are only considering a single vectorcomponent x, vx, ax it is easier to represent it asa function of time, e.g. position-versus-time graph:

Δt = 1 min

Plot x as a function of tinstead of correspondingmotion diagram:

y[m]

x [m]200 400 600

t1 t4 t7 t10

“y versus x” graph


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→

→

Interpreting a position (or x versus t) graph

As an example, reflective of many problems you will find in your textbook, interpret the one dimensional motion associated with the position graph given below

This represents the motionof a car along a straight road.


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1.7a From words to symbols


1.7b A problem solving strategy

1.7 Solving Problems in Physics p. 20-24

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→

→

Much of physics is FIRST about thinking (understandingthe question/problem), SECOND is reasoning (leading tothe pictorial/physical representation; i.e. motion diagram)then, THIRD is using/solving the equations once youhave isolated the appropriate ones.

Slightly re-worded from Tactic box 1.5 (page 21 In your textbook):

Sketch the situation. Show the object at the beginning, end, and at any pointwhere the character of the motion changes. Simple drawings are adequate. Establish a coordinate system. Select your axes and origin. Define symbols. Use the sketch to define symbols representing quantitiessuch as position, velocity, acceleration, and time. Every variable used later in themathematical solution should be defined on the sketch. List known information. Make a table of the quantities whose values you candetermine from the problem statement or that can be found quickly. Identify the desired unknowns. What quantities will allow you to answer thequestion? Don’t list every unknown; only those needed for the answer.

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→

→

A good problem-solving strategy is based onvisualizing a problem. This will make theMath much easier later on.

Model: It’s 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 where expert problem solvers put most of their effort.

. Draw a pictorial representation. This helps you assess the information you are given and starts the process of translating the problem into symbols.. Draw a physical representation. This helps you visualize important aspects of the physics. Motion diagrams are part of the physical representation. Chapter 4 will introduce free-body diagrams to display information about forces.. Use a graphical representation if it is appropriate for the problem.. Go back and forth between these three representations; they need not be done in any particular order.

Solve: Only after modeling and visualizing are complete is it time to develop a mathematical representation with specific equations that must be solved. All symbols used here should have been defined in the pictorial representation.

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

(See STRATEGY BOXIn Page 23)

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1.8 Units and Significant figures p. 24-28


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→

→

Units are an extremely important concept ofscience in general

All what we can describe in our world is based onmeasurements; there is nothing “outside” of theuniverse which could provide us with a universallength, for instance

Therefore, units are nowadays defined by themost precise experiments available

Some definitions therefore sound somewhatsilly

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→

→

For instance, one SECOND is defined as9,192,631,770 times the oscillation between thetwo hyperfine levels of the cesium-133 atom(133Cs) the atomic clock!

Only the KILOGRAM is still defined by an object(and that object looses weight because it wastoo often polished) and not by a preciseexperiment, but that may change soon.

One METER is the distance travelled by light inVacuum during 1/(299,792,458) of a second.

This number, close to three hundred millions, isthe speed of light (299,792,458 m/s)Ph

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→

→

Not using SI units may end in a disaster, likethe US mars polar lander (MPL) crash in 1999,

http://news.bbc.co.uk/1/hi/sci/tech/4522291.stm

“The board that examinedits loss concluded thecause lay in a mix-up overunits, with one team on themission using metricmeasurements andanother using English units(US equivalent of UKimperial).”

It is also important to agree upon units. We willalways use SI (Systeme International) units

second, meter, kilogram

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→

→

When calculating a quantity, make sure thatyou ALWAYS give the correct SI unit, e.g.,

ATTENTION: Using the wrong/incorrect units in a test or assignment WILL cost you marks!

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Unit Conversions


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Unit conversion is usually a simple matter ofexpressing one unit through another:

40


→

→

Estimating numbers is very important to checkif your results are meaningful

Keep [ 1 m/s = 3.6 km/h ] in mind whenestimating (or working with) velocities

Keep [ g ≈ 10 m/s2 ] in mind when estimating(or working) with acceleration

Also have a look at Table 1.5 and 1.6 in the book,it will help you to get a feeling for numbers

If your calculations show the acceleration of gravityon the moon to be g = 35 m/s , the chances are you have made a mistake.

2

If your calculations show the speed of the C-Trainto be v = 100 m/s as it approaches downtown, the chances are you have made a mistake.

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Significant figures (in General)

Multiplying two numbers:

Adding two numbers:


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→

→

E.g. 4.21 / 7.32 = 0.575137 is nonsense

Since input accuracy is only three digits, therefore

4.21 / 7.32 = 0.575 is appropriate

Another example for significant figures if inputaccuracy is three digits: 0.00620 = 6.20 × 10-3 is appropriate

Rule1: don't write down the 8 digits your calculator displays. Rule2: see rule number one.

When doing a precise calculation (rather than anestimation) only calculate significant figures.

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Many calculators have a default setting that shows 2decimal places, such as 5.23. THIS IS MISLEADING!!!

Say you are trying to calculate 5.23/58.5 to within 3significant digits. Your calculator gives 0.09 while the trueanswer is

0.0894 or 8.94x10

IMPORTANT!

You will avoid this error if you keep your calculator set to display numbersIn SCIENTIFIC NOTATION with 2 decimal places.

-2

If you enter this as the answer you will lose marks!

Know your calculator!


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…. Scientific Notation


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Scientific Notation (continued)


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→

→

Another important part of scientific language(that is all sciences): prefixes

Common prefixes include

Giga G 1 GV = 109 Volt Mega M 1 Mt = 106 tons kilo k 1 km = 103 meter milli m 1 ml = 10-3 liter micro µ 1 µσ = 10-6 seconds nano n 1 nm = 10-9 m

No way around, you need to memorize this

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“CONCEPTS” QUIZES

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What is a “particle”?

1. Any part of an atom2. An object that can be represented as a mass at a single

point in space3. A part of a whole4. An object that can be represented as a single point in

time5. An object that has no top or bottom, no front or back

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What is a “particle”?

1. Any part of an atom2. An object that can be represented as a mass at a

single point in space3. A part of a whole4. An object that can be represented as a single point in

time5. An object that has no top or bottom, no front or back

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What quantities are shown on a complete motiondiagram?

1. The position of the object in each frame of the film,shown as a dot

2. The average velocity vectors (found by connectingeach dot in the motion diagram to the next with avector arrow)

3. The average acceleration vectors (there is oneacceleration vector linking each two velocity vectors)

4. All of the above

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What quantities are shown on a complete motiondiagram?

1. The position of the object in each frame of the film,shown as a dot

2. The average velocity vectors (found by connectingeach dot in the motion diagram to the next with avector arrow)

3. The average acceleration vectors (there is oneacceleration vector linking each two velocity vectors)

4. All of the above

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An acceleration vector

1. tells you how fast an object is going.2. is constructed from two velocity vectors.3. is the second derivative of the position.4. is parallel or opposite to the direction of motion.5. Acceleration vectors weren’t discussed in this chapter.

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An acceleration vector

1. tells you how fast an object is going.2. is constructed from two velocity vectors.3. is the second derivative of the position.4. is parallel or opposite to the direction of motion.5. Acceleration vectors weren’t discussed in this chapter.

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The pictorial representation of a physics problemconsists of

1. a sketch.2. a coordinate system.3. symbols.4. a table of values.5. all of the above.

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The pictorial representation of a physics problemconsists of

1. a sketch.2. a coordinate system.3. symbols.4. a table of values.5. all of the above.

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Chapter 1 (Clictions)


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Checking Understanding

Maria is at position x = 23 m. She then undergoes a displacement∆x = –50 m. What is her final position?

A. –27 m

B. –50 m

C. 23 m

D. 73 m


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Maria is at position x = 23 m. She then undergoes a displacement∆x = –50 m. What is her final position?

A. –27 m

Answer


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Two runners jog along a track. The positions are shown at 1 stime intervals. Which runner is moving faster?


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Two runners jog along a track. The positions are shown at 1 stime intervals. Which runner is moving faster?

Answer


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Two runners jog along a track. The times at each position areshown. Which runner is moving faster?


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Two runners jog along a track. The times at each position areshown. Which runner is moving faster?

They are both moving at the same speed.


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Person runs from x1 = 50.0 m to x2 = 30.5 min ∆t = 3.0 s.

Displacement Average Velocity A) 19.5 m 6.5 m/s B) -19.5 m -6.5 m/s C) -80.5 m -20.5 m/s D) 80.5 20.5 m/s


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Person runs from x1 = 50.0 m to x2 = 30.5 min ∆t = 3.0 s. ∆x = -19.5 m

Average velocity = (∆x)/(∆t)= -(19.5 m)/(3.0 s) = -6.5 m/s. Negative signindicates DIRECTION, (negative x direction)


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Average Velocity&

Average SpeedA person walks 70 m east, then 30 m west in a total of 100 seconds.

Displacement Speed Average Velocity

a) A) 100 m 2.2 m/s -0.57 m/sb) B) 40 m 1.4 m/s -0.57 m/sc) C) 40 m 1.4 m/s 0.57 m/sd) D) -40 m 1.4 m/s 0.57 m/se) E) 40 m 0.57 m/s 1.4 m/s


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Walk for 70 s.

Average Speed = (100 m)/(70 s) = 1.4 m/s Average velocity = (40 m)/(70 s) = 0.57 m/s

C


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Acceleration is:

A) + 5 m/s2

B) -5 m/s2

C) 0 m/s2

D) 2.0 m/s2

E) - 2 m/s2


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E


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The same car as in before but now moving to theleft and decelerating.

Acceleration is:

A) + 5 m/s2

B) -5 m/s2

C) 0 m/s2

D) 2.0 m/s2

E) - 2 m/s2


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“Deceleration”: A word meaning “slowing down”. We try to avoid using itin physics. Instead (in one dimen.) talk about positive & negativeacceleration.

This is because (for one dimen. motion) deceleration does notnecessarily mean the acceleration is negative!

Deceleration !!

2 D


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Selected Problems

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End of Chapter 1

IMPORTANT:

Print a copy of the SUMMARY page (p. 29)and add it here to your lecture notes.

It will save you crucial time when trying to recall:Concepts, Symbols, and Strategies

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chapter 1: concepts of motion - capca€¦ · 2 chapter 1: concepts of motion 1.1 motion diagrams...

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