A Mathematical Model of Motion

CHAPTER 5

PHYSICS

5.1 Graphing Motion in One Dimension Interpret graphs of position versus time for

a moving object to determine the velocity of the object

Describe in words the information presented in graphs and draw graphs from descriptions of motion

Write equations that describe the position of an object moving at constant velocity

Parts of a GraphX-axisY-axisAll axes must be labeled with

appropriate units, and values.

5.1 Position vs. Time The x-axis is always

“time” The y-axis is always

“position” The slope of the line

indicates the velocity of the object.

Slope = (y2-y1)/(x2-x1) d1-d0/t1-t0

Δd/Δt

Position vs. Time

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Pos

itio

n (m

)

Uniform Motion

Uniform motion is defined as equal displacements occurring during successive equal time periods (sometimes called constant velocity)

Straight lines on position-time graphs mean uniform motion.

Given below is a diagram of a ball rolling along a table. Strobe pictures reveal the position of the object at regular intervals of time, in this case, once each 0.1 seconds. 

Notice that the ball covers an equal distance between flashes. Let's assume this distance equals 20 cm and display the ball's behavior on a graph plotting its x-position versus time.

The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This represents the ball's average velocity as it moves across the table. Since the ball is moving in a positive direction its velocity is positive. That is, the ball's velocity is a vector quantity possessing both magnitude (200 cm/sec) and direction (positive).

Steepness of slope on Position-Time graph

Slope is related to velocity Steep slope = higher

velocityShallow slope = less

velocity

Different Position. Vs. Time graphsPosition vs. Time

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Pos

itio

n (m

)

Position vs. Time

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

Time (s)

Pos

itio

n (m

)

Position vs. Time

0

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4

6

810

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Time (s)

Position vs. Time

0

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Time (s)

Constant positive velocity(zero acceleration)

Constant negative velocity(zero acceleration)

Increasing positive velocity(positive acceleration)

Decreasing negative velocity(positive acceleration)

Uniform Motion Accelerated Motion

Different Position. Vs. Time

Position vs. Time

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1 2 3 4 5

Time (s)

Position vs. Time

0

2

4

6

8

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Time (s)

Changing slope means changing velocity!!!!!!

Decreasing negative slope = ?? Increasing negative slope = ??

X

t

A

B

C

A … Starts at home (origin) and goes forward slowlyB … Not moving (position remains constant as time progresses)C … Turns around and goes in the other direction quickly, passing up home

During which intervals was he traveling in a positive direction?During which intervals was he traveling in a negative direction?During which interval was he resting in a negative location?During which interval was he resting in a positive location?During which two intervals did he travel at the same speed?A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

Graphing w/ Acceleration

x

A … Start from rest south of home; increase speed gradually

B … Pass home; gradually slow to a stop (still moving north)

C … Turn around; gradually speed back up again heading south

D … Continue heading south; gradually slow to a stop near the starting point

t

A

B C

D

You try it…..

Using the Position-time graph given to you, write a one or two paragraph “story” that describes the motion illustrated.

You need to describe the specific motion for each of the steps (a-f)

You will be graded upon your ability to correctly describe the motion for each step.

Tangent Lines

t

SLOPE VELOCITY

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

Increasing & Decreasing

t

x

Increasing

Decreasing

On a position vs. time graph:

Increasing means moving forward (positive direction).

Decreasing means moving backwards (negative direction).

Concavityt

x

On a position vs. time graph:

Concave up means positive acceleration.

Concave down means negative acceleration.

Special Points

t

x

PQ

R

Inflection Pt. P, RChange of concavity,

change of acceleration

Peak or Valley

QTurning point, change of

positive velocity to negative

Time Axis Intercept

P, STimes when you are at

“home”, or at origin

S

5.2 Graphing Velocity in One Dimension Determine, from a graph of velocity versus

time, the velocity of an object at a specific time

Interpret a v-t graph to find the time at which an object has a specific velocity

Calculate the displacement of an object from the area under a v-t graph

5.2 Velocity vs. Time

X-axis is the “time”

Y-axis is the “velocity”

Slope of the line = the acceleration

Velocity vs. Time

0

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Time (s)

Vel

coit

y (m

/s)

Different Velocity-time graphs

Velocity vs. Time

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Time (s)

Velcoity vs. Time

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Time (s)

Different Velocity-time graphsVelocity vs. Time

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Time (s)

Vel

ocit

y (m

/s)

Velocity vs. Time

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Time (s)

Vel

ocit

y (m

/s)

Velocity vs. Time

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Velocity vs. Time

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Velocity vs. Time Horizontal lines = constant velocity Sloped line = changing velocity

Steeper = greater change in velocity per second

Negative slope = deceleration

Acceleration vs. Time

Time is on the x-axis Acceleration is on the

y-axis Shows how

acceleration changes over a period of time.

Often a horizontal line.

Acceleration vs. Time

0

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Time (s)

Acc

eler

atio

n (m

/s^

2)

All 3 Graphst

x

v

t

a

t

Real life

a

t

v

t

Note how the v graph is pointy and the a graph skips. In real life, the blue points would be smooth curves and the orange segments would be connected. In our class, however, we’ll only deal with constant acceleration.

Constant Rightward Velocity

Constant Leftward Velocity

Constant Rightward Acceleration

Constant Leftward Acceleration

Leftward Velocity with Rightward Acceleration

Graph PracticeTry making all three graphs for the following scenario:

1. Newberry starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed.

2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose.

3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail.

4. Fleeing the scene of the crime, Newberry takes off again in the same direction, speeding up quickly.

5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.

Area Underneath v-t Graph

If you calculate the area underneath a v-t graph, you would multiply height X width.

Because height is actually velocity and width is actually time, area underneath the graph is equal to

Velocity X time or

V·t

Remember that Velocity = Δd Δt

Rearranging, we get Δd = velocity X Δt

So….the area underneath a velocity-time graph is equal to the displacement during that time period.

Areav

t

“positive area”

“negative area”

Note that, here, the areas are about equal, so even though a significant distance may have been covered, the displacement is about zero, meaning the stopping point was near the starting point. The position graph shows this as well.

t

x

Velocity vs. Time The area under a velocity time graph indicates

the displacement during that time period. Remember that the slope of the line indicates

the acceleration. The smaller the time units the more

“instantaneous” is the acceleration at that particular time.

If velocity is not uniform, or is changing, the acceleration will be changing, and cannot be determined “for an instant”, so you can only find average acceleration

5.3 Acceleration

Determine from the curves on a velocity-time graph both the constant and instantaneous acceleration

Determine the sign of acceleration using a v-t graph and a motion diagram

Calculate the velocity and the displacement of an object undergoing constant acceleration

5.3 Acceleration

Like speed or velocity, acceleration is a rate of change, defined as the rate of change of velocity

Average Acceleration = change in velocity t

VVa

01

Elapsed time Units of acceleration?

Rearrangement of the equation

t

VVa

01

10 vtav

01 vvta

atvv 01

Finally…

This equation is to be used to find (final) velocity of an accelerating object. You can use it if there is or is not a beginning velocity

atvv 01

Displacement under Constant Acceleration

Remember that displacement under constant velocity was

With acceleration, there is no one single instantaneous v to use, but we could use an average velocity of an accelerating object.

Δd = vt or d1 = d0 + vt

Average velocity of an accelerating object

V = ½ (v0 + v1)

Average velocity of an accelerating object would simply be ½ of sum of beginning and ending velocities

So…….

tvvdd

vtdd

)(2/1 0101

01

tvvdd )(2/1 0101

Key equation

Some other equations

attvdd 2/1001 2

This equation is to be used to find final position when there is an initial velocity, but velocity at time t1 is not known.

If no time is known, use this to find final position….

a

vvdd

2

0101

22

Finding final velocity if no time is known…

)(2 0101 ddavv 2 2

The equations of importance

t

VVa

01

atvv 01

tvvdd )(2/1 010

attvdd 2/100

a

vvdd

2

0101

)(2 0101 ddavv

2

2 2

2 2

Practical Application Velocity/Position/Time equations Calculation of arrival times/schedules of aircraft/trains

(including vectors) GPS technology (arrival time of signal/distance to

satellite) Military targeting/delivery Calculation of Mass movement (glaciers/faults) Ultrasound (speed of sound) (babies/concrete/metals)

Sonar (Sound Navigation and Ranging) Auto accident reconstruction Explosives (rate of burn/expansion rates/timing with det.

cord)

5.4 Free Fall

Recognize the meaning of the acceleration due to gravity

Define the magnitude of the acceleration due to gravity as a positive quantity and determine the sign of the acceleration relative to the chosen coordinate system

Use the motion equations to solve problems involving freely falling objects

Freefall

Defined as the motion of an object if the only force acting on it is gravity.No friction, no air resistance, no drag

Acceleration Due to Gravity

Galileo Galilei recognized about 400 years ago that, to understand the motion of falling objects, the effects of air or water would have to be ignored.

As a result, we will investigate falling, but only as a result of one force, gravity. Galileo Galilei 1564-1642

Galileo’s Ramps

Because gravity causes objects to move very fast, and because the time-keepers available to Galileo were limited, Galileo used ramps with moveable bells to “slow down” falling objects for accurate timing.

Galileo’s Ramps

Galileo’s Ramps

To keep “accurate” time, Galileo used a water clock. For the measurement of time, he employed a large

vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which he collected in a small glass during the time of each descent... the water thus collected was weighed, after each descent, on a very accurate balance; the difference and ratios of these weights gave us the differences and ratios of the times...

Displacements of Falling Objects

Looking at his results, Galileo realized that a falling ( or rolling downhill) object would have displacements that increased as a function of the square of the time, or t2

Another way to look at it, the velocity of an object increased as a function of the square of time, multiplied by some constant.

Galileo also found that all objects, no matter what slope of ramp he rolled them down, and as long as the ramps were all the same height, would reach the bottom with the same velocity.

Galileo’s Finding

Galileo found that, neglecting friction, all freely falling objects have the same acceleration.

Hippo & Ping Pong Ball In a vacuum, all bodies fall at the same rate.

When there’s no air resistance, size and shape matter not!

If a hippo and a ping pong ball were dropped from a helicopter in a vacuum (assuming the copter could fly without air), they’d land at the same time.

Proving Galileo CorrectGalileo could not produce a vacuum to prove his ideas. That came later with the invention of a vacuum machine, and the demonstration with a guinea feather and gold coin dropped in a vacuum.

Guinea Feather and Coin/NASA demonstrations

Acceleration Due to Gravity

Galileo calculated that all freely falling objects accelerate at a rate of

9.8 m/s2

This value, as an acceleration, is known as g

Acceleration Due to Gravity

Because this value is an acceleration value, it can be used to calculate displacements or velocities using the acceleration equations learned earlier. Just substitute g for the a

Example problem

A brick is dropped from a high building. What is it’s velocity after 4.0 sec.?How far does the brick fall during this time?

The Church’s opposition to new thought Church leaders of the time held the same views

as Aristotle, the great philosopher. Aristotle thought that objects of different mass

would fall at different rates…makes sense huh??????

All objects have their “natural position”. Rocks fall faster than feathers, so it only made sense (to him)

Galileo, in attempting to convince church leaders that the “natural position” view was incorrect, considered two rocks of different mass.

Falling Rock Conundrum

Galileo presented this in his book Dialogue Concerning the Two Chief World Systems(1632) as a discussion between Simplicio (as played by a church leader) and Salviati (as played by Galileo)

Two rocks of different masses are dropped Massive rock falls faster???

Rocks continued

Now consider the two rocks held together by a length of string.

If the smaller rock were to fall slower, it would impede the rate at which both rocks would fall.

But the two rocks together would actually have more mass, and should therefore fall faster.

A conundrum????? The previously held views could not have been correct.

Galileo had data which proved Aristotle and the church wrong, but church leaders were hardly moved in their position that all objects have their “correct position in the world”

Furthermore, the use of Simplicio (or simpleton) as the head of the church in his dialog, was a direct insult to the church leaders themselves.