chapter 2: kinematics, the math of motion -...

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2.1 Uniform Motion pp. 34-382.1 Uniform Motion pp. 34-382.2 Instantaneous velocity pp. 38-442.2 Instantaneous velocity pp. 38-442.3 Finding position from velocity pp. 44-482.3 Finding position from velocity pp. 44-482.4 Motion with constant acceleration pp. 48-542.5 Free Fall pp. 54-572.6 Motion on an inclined plane pp. 57-612.7 Instantaneous acceleration pp. 61-62

Chapter 2: Kinematics, the math of motionPh

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2.1 Uniform motion p. 34-38


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Chapter 2: Kinematics, the math of motion

→

The simplest form of motion is uniform motion

A motion is uniform if its position-versus-timegraph is a straight line

Δt

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For uniform motion the (average) velocity remainsconstant:

An alternative way to say “uniform motion is de-scribed by a straight line in the position-versus-time graph”, is to say that for uniform motion x(t) isa linear function of t:

x(t) = x(0) + vx t NOTE:x(0) is x(t=0)

This means “x” as a functionof time “t” NOT “x” times “t”as in a(b+c)=ab+ac!

Note the units:

MetersSeconds

= m/s

Recall: the equation of a straight line is y(x) = a x + b

slope y-interceptPhy2

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This is even true for uniform motion in a general direction (say s; not necessarily along the x axis)

Example; along an inclined plane!

Ball rolling down the inclined plane

X-axis

Y-axis

X3 X2

t1

Y1

Y2

Y3

I can define a new variable

s=(X,Y) such that s = (X ,Y ); s = (X , Y ); s = (X , Y ) .. etc …

so that the distance of the object from the origin of my co-ordinate system is

s =

(0,0)

3 3 322111 2

√x2+ y2

X1

t2t3

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If si [meaning (xi,yi)] is the initial component of theposition vector in the s-direction, and sf [meaning(xf,yf)] its final position after uniform motion for atime Δt, then

with Δt = tf-ti and vs the velocity component along the s axis

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Cliction 2.1

Which position-versus-time graph represents themotion shown in the motion diagram below?

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Tactic box 2.1 (p.36): Interpreting position-versus-time graphs

Steeper slopes correspond to faster speeds

Negative slopes correspond to negative velocities, i.e., to motion to the left (or down)

The slope is a ratio of intervals, not a ratio of coordinates. That is, the slope is NOT simply x/t (x divided by t)

Note that position-versus-time graphs have units on their axes.

The “rise” Δx is some number of meters; the “run” Δt is some number of seconds. The physically meaningful rise and run include units, and the ratio of these units gives the units of the slope, i.e., the velocity.

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Practice Problem(use units of miles and hours)

Susan’s speed and velocity

Bob’s speed and velocity

The equation that describesSusan’s position-versus-time (in this case, x-vs-t) graph of motion.

The equation that describes Bob’s position-versus-time (in this case, x-vs-t) graph of motion

Finally, use (it means equate) the two equations to find the exact co-ordinates of the point of meeting.

Recall: the equation of a straight line is y(x) = a x + b

slope The y-intercept when x=0Phy2

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2.2 Instantaneous velocity p. 38-44


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Suppose a physicist (disguised as a police officer) pulls you over and says,

“I just clocked you going 80 miles per hour”.

You might respond,

“But that’s impossible. I have only been driving for 20 minutes, so I can’t possibly have gone 80 Miles.”

He replies (and that’s when you suspect that you are dealing with a physicist)

“I mean at the instant I measured your velocity, you were moving at rate such that youWOULD cover a distance of 80 miles IF you were to continue at that velocity withoutchange for an hour.”

You try to be smart:

“The law does not specify if the limit applies for AVERAGE or INSTANTANEOUS velocity”

The police who has obviously taken PHYS221 adds

“That will be an INSTANT $200 fine!”

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A very important notion is that of instantaneousquantities like the instantaneous velocity

We defined the velocity as vs = Δs/Δt, butthis is only the average velocity during Δt

To get the instantaneous velocity let the timeinterval between two positions decrease, or Δt 0

t [s]

x [m]

1

2

t [s]

x [m]

12

Δt Δt t [s]

x [m]

12

Δt → 0

This meansit approaches

Zero but NEVER

goes to zero

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→

Δt → 0t [s]

x [m]

1

2

t [s]

x [m]

12

Δt Δt t [s]

x [m]

12

As Δt → 0 we also have Δs → 0

0

0=

We have a dilema!Phy2

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The way out of this dilemma is to use limits:

This means the instantaneous velocity at time t is thederivative of the position s(t) with respect to time t.

This means as time interval approaches zero!

Object location at tObject location at t+ Δt

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Cliction 2.2

Which velocity-versus-time graph goes withthis position-versus-time graph on the left?

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Before applying what we have learned about theinstantaneous velocity, we should first do someCALCULUS involving DERIVATIVES:

PRACTICE using Textbook examples 2.5 and 2.6!

This is a “dummy name”. In other words we could have used f(t) or x(t)!

“c” and “n” are constants; meaning fixed numbers

The derivative of a sumIs the sum of the

Derivatives!

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Practice Problem (use SI units)

You are given the position-versus-time graphusing: s(t) = 2t m

Derive the corresponding velocity-versus-time graph.

2

u=s c=2

n=2

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Example:

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Example:

Motion diagram:

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Example:

Motion diagram:

Turning points

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Cliction

1. The slope at a point on a position-versus-time graph of anobject isA. the object’s speed at that point.B. the object’s average velocity at that point.C. the object’s instantaneous velocity at that point.D. the object’s acceleration at that point.E. the distance traveled by the object to that point.


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Cliction

2. The area under a velocity-versus-time graph of an object isA. the object’s speed at that point.B. the object’s acceleration at that point.C. the distance traveled by the object.D. the displacement of the object.E. This topic was not covered in this chapter.


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End of Week 2


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2.3 Finding position from velocity p. 44-48


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→

We are now dealing with the reverse problem:finding the position s(t) of a particle when we knowits velocity vs(t). In other words,

This integral is the area between the function vs(t) and the t-axis.

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This symbol means ADD all of the area of RECTANGLES as you go from t to t .

i f

Integral = Sum(Rectangles)=Sum(Δt.v )i

NOTE:The units of Δt.v are

[seconds].[meters/seconds]which yields

[meters]

i

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We should now recall some CALCULUS involvingINTEGRALS (inverse operation of differentiation):This is a definite integral because there are two definite boundariesff the area we want to find.

The Integral of a sumis the sum of the

Integrals!

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This is the velocity-versus-timediagram of a drag racer.

Given the velocity-versus-time graph above, whatis the position-versus time graph?

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Solution:

QUESTIONHow far does the racer moveduring the first 3 seconds?

X(3) = 2.3 = 2.9 = 18 meters.2

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Cliction 2.3

Which position-versus-time graph goes with this velocity-versus-time graph on the left? The particle’s position atti = 0 s is xi = –10 m .

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2.4 Motion with constant acceleration p. 48-54


To say that “an object has a uniform acceleration of 3 m/s”

is equivalent to saying that “its velocity increases by 3 m/s every second”.

2

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We now consider the second (of two) important types ofmotion: motion with constant accelerationRemember that acceleration is the change of the velocityduring a time interval Δt,

→An object whose velocity-versus-time graph is a STRAIGHT LINE has

a steady and unchanging acceleration.

For a uniformly accelerated motion,the SLOPE of the velocity-versus-time

graph (that is the acceleration) is CONSTANT!

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→

If a is constant then the velocity is a linear functionof time:→

The area under the straight line can be divided into

rectangles. Adding the areas of these

give you the object’s velocity in time!

Final velocity

Initial velocity

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This is probably the most important equation inPhys 221. Hints to remember it: vx varies linearly its integral x varies quadrat. Check the units: to get a length, vx needs to be multiplied with Δt, ax with Δt2 ax/2 so that it becomes ax after differentiation

The shaded area can be dividedinto rectangles and triangles. Adding the areas of these give

you the object’s position in time!

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Put a = 0 in these equations

Keep a=constant in these equations

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Cliction 2.4

Which velocity-versus-time graph or graphs goes withthis acceleration-versus-time graph? The particle isinitially moving to the right and eventually to the left.

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2.5 Free Fall p. 54-57


EXPERIMENT

Hold your phys221 text book (or any book) in one hand and a sheet of paper of similar size (or smaller) in the other hand.

Drop them from the same height and at the same time. Which one reaches the floor first?

Then, hold the same book but now put the sheet of paper on top of it, and drop them from the same height as before.

Which one reaches the floor first?

WHAT DO YOU CONCLUDE ?

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Free fall is the first natural constant accelerationcase to study:

If air resistance can be neglected, and if we stayrelatively close (50km) to the ground, all objectsexperience the same constant acceleration andpoints downward:

afree fall = (9.80 m/s2, vertically downward) = (0, -9.80 m/s2 )

or ay = -9.80 m/s2 = -g

→

g is NOT “gravity”!g is “the acceleration

due to gravity”

IMPORTANT!!!

IMPORTANT!!!

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Example: How long does a falling rock takes to reachthe ground when starting to fall at y0 = 100m

NOTE:V (0) = V = 0 m/sy 0y

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2.6 Motion on an inclined plane p. 57-61


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Motion on an inclined plane is also an example ofuniformly accelerated motion

The tricky part is to identify the relevantcomponent of the Earth's acceleration

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Trick to simplify description: choose suitablecoordinate system

x

y We then only need toconsider x(t)

Components of accel-eration in the tilted co-ordinate system:

One-Dimensional Motion

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The inclined plane prevents the object to move iny-direction, we therefore have y(t)=0 for all times

The motion along thex-axis is now subjectto the constant accele-ration ax = g sin( )

⇒

x

y

!

"

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Cliction 2.5

The ball rolls up the ramp, then back down. Which isthe correct acceleration graph?

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→

This is the table listing theknown quantities.

An amusement park shoots a car up a frictionless track inclined at 30 . The car rolls up,

then rolls back down. If the height of the track is 20 m, what is the maximum allowable

speed with which the car can start?


o

Symbol for degrees

First figure out the sign of the acceleration a.Then, figure out which one of the 3 equations

listed in table above is applicable here.(see detailed solution in page 63/64

in Textbbok)

s

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→

The one-dimensional acceleration along an incline is

The CORRECT SIGN depends on the direction in which the ramp is tilted!

IMPORTANT!

x

x !

a = ±gsin(")x

!

a = +gsin(")

!

a = "gsin(#)

NOTEg is the length, or magnitude,

of the acceleration due togravity

g = 9.80 m/s2

x x

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→

IMPORTANT!

x

x !

a = +gsin(")

!

a = "gsin(#)xf

!

V =V + 2a"x2 2

i

2

x

!

V =V + 2a"x2

f i

!

"initial

final

h

!

"initial

final

h

!

"x = X - X > 0final initial

!

"x = X - X < 0final initial

x x

V = 0f V = 0f

!

V = "2a#xxi2

> 0

!

V = "2a#xxi2

> 0

< 0 > 0

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2.7 Instantaneous acceleration p. 61-62


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→

Also useful to understand the concepts of motionis: instantaneous accelerationThis can be introduced in exactly the same way asinstantaneous velocity: ax is the change ofvx during a time interval Δt.

Let Δt go to zero

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→

IMPORTANTComparing instantaneous velocity with instantaneous acceleration

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Cliction 2.6

Rank in order, from largest to smallest (amplitude),the accelerations aA– aC at points A – C.

1) aA > aB > aC2) aC > aA > aB3) aC > aB > aA4) aB > aA > aC

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Example: encounter of two objects

A BMW starts at xBMW(0) = 0m (we set the origin atx=0m,t=0s) and accelerates with ax=g. When doesit overtake a light beam that started at the sameposition and has constant velocity c = 2.99x108

m/s? Is this reasonable?Constant velocity (or a = 0)

s = s( t ) + V . (t - t )0 0

s = 0 + V . (t - 0)

s = V . t

s

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Solution:

The solution is not reasonablesince the final velocity of theBMW is vf=g tenc = 2 c, butnothing can go faster than light.

x [m]

xBMW(t)

xLight(t)

t[s]tenc

x=c.tLight

x= g.t21 2BMW

x= g.t21 2

BMW v= g.t

X(t) = x(t ) + v(t ).(t - t ) + g (t - t )0 0 00

212

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→

Let's apply all we learned so far to a real problem:a 25 m ski jump.

skijump.wmv

View movie at:

www.capca.ucalgary.ca/~rouyed/teaching/Mechanics/phys221.html

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→

motion diagram y-position-time graph

25

x[m]

y[m]

0

t[s]

y[m]

0

t=2st=0s

t=4s

t=6s 42

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Translate this into Mathematics:

Stage 1: rocky track,let's assume the mo-tion is described by

for 0<t<2, t in seconds,y in meter.

t[s]

y[m]

042

Stage 1:non-uniformmotion

Stage 2:free fall

Stage 3:inclined plane

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→

What is his velocity at t=2s?

vy(t=2s) = 0 m/s

This determines the initial conditions for stage 2

t[s]

y[m]

042


Stage 2:free fall


How high is he at t=2s? Answer: y(t=2s) = 25 m

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→

Stage 2: free fall. Motion is described by

with t0 = 2s the start offree fall. We knowy(t0)= 25m and vy(t0) =0 m/s

at t=4s (with g≈10 m/s2)we find y(t=4s) ≈ 5 m t[s]

y[m]

042


Stage 2:free fall


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Stage 3: inclined plane. Let's assume the snowstopped him so that his initial speed in the planeis 1m/s. The angle is 45°so that along the direction of the plane

with t0=4s the start ofthe motion on the plane.The snow stopped himso that vs(t0)=1m/s

t[s]

y[m]

042


Stage 2:free fall


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→

How far has he travelled on the plane after t=6s?

How fast is he then?

t[s]

y[m]

042


Stage 2:free fall


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Reading QuizChapter 2


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The slope at a point on a position-versus-time graph ofan object is

1) the object’s speed at that point.2) the object’s average velocity at that point.3) the object’s instantaneous velocity at that point.4) the object’s acceleration at that point.5) the distance traveled by the object to that point.


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The area under a velocity-versus-time graphof an object is

1) the object’s speed at that point.2) the object’s acceleration at that point.3) the distance traveled by the object.4) the displacement of the object.5) This topic was not covered in this chapter.


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At the turning point of an object,

1) the instantaneous velocity is zero.2) the acceleration is zero.3) Both 1 and 2.4) Neither 1 nor 2.5) This topic was not covered in this chapter.


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A 1-pound round ball and a 100-pound round ball areplaced side-by-side at the top of a frictionless hill. Eachis given a very light tap to begin their race to the bottomof the hill. In the absence of air resistance

1) the 1-pound ball wins the race.2) the 100-pound ball wins the race.3) the two balls end in a tie.4) there’s not enough information to determinewhich ball wins the race.


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Selected Problems

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End of Chapter 2

IMPORTANT:

Print a copy of the SUMMARY page (p. 63)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 2: kinematics, the math of motion -...

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