Guided Notes – Pendulums



1.1.1 Displacement and Distance

Definitions

• _______________________ - a measured quantity that has NO direction

•

• _______________________ - a measured quantity that INCLUDES direction (INDICATED BY SIGN!)

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Sign Conventions

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Example #1

A man drives his car 3 miles north, then 4 miles east.

What distance did he travel?

What is his displacement from his point of origin?

Example #2

Three men leave the same house on foot. The first man walks 30 feet north, then 40 feet west. The second man walks 90 feet south, then 88 feet north. The third man walks 10 feet east, then 50 feet west.

Which man has traveled the greatest distance? ______

Who is farthest from the house? _______

Who is closest to the house? _______

Distance vs. Time Graph

During what time interval is the object not moving? ______________

Displacement vs. Time Graph

During what time interval(s) is the object not moving? ______________________

At what distance from the origin did the object stop? _____________________

During what time interval(s) was the object to the left of the origin? _________________

PRACTICE

1) Answer the following questions using the scaled diagram shown below.

a. Wendy swims from point A on the west bank of the stream to point B on the east bank. Using a ruler, measure the distance that she swims.

b. What is her displacement from her origin at point A?

c. Jake starts walking from point C and uses the bridge to cross the stream, then turns and walks along the stream to point B. Use a ruler to determine the distance that Jake walks.

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1.1.2 Velocity and Speed

Definitions

• ________________: change in displacement occurring over time

– Includes both _______________ and ______________

– ________________

• _________________: change in distance occurring over time

– Includes _____________________ only!

– ________________

Average Velocity/Speed Equations

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Example #1

Jim gets on his bike and rides 300 meters west in 60 seconds.

– What is his average velocity?

– What is his average speed?

Example #2

Sally gets up one morning and decides to take a 3.0 mile walk. She completes the first mile in 8.0 minutes, the second mile in 8.5 minutes and the third mile in 9.0 minutes.

– What is her average speed during her walk?

Example #3

A car’s speed changes from 20 meters per second to 10 meters per second.

– What is her average speed during her walk?

Graphs

What can we say about the velocity of the objects in each of the graphs below?

Uses for a v-t graph

PRACTICE

1) Jane rides her bike a distance of 3500 meters with an average speed of 6.8 meters per second. Calculate the time for which Jane rode.

2) A shark swims with an average speed of 10 meters per second for 2 hours. Calculate the distance that the shark travels in this time.

3) A rabbit comes out of its hole at point A just as a bobcat arrives at the same point. The rabbit then runs due west for 20 meters, then turns due north and runs for 30 meters (arriving at point B). It takes the rabbit 5 seconds to complete this trip. The bobcat begins running at the same time as the rabbit, but heads directly for point B without making any turns.

a. What is the rabbit’s average speed? What is its average velocity?

b. How fast must the bobcat run in order to reach point B at the same time as the rabbit?

Graphing Practice

Graph displacement vs. time and velocity vs. time for each of the following.

1) A ball sits motionless on a table at a position 3 meters to the right of the origin.

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2) A man starts at a location 5 meters north of his house. He stands there for 3 seconds then walks away from his house with a constant speed of 1 meter per second.

[pic]

3) A boy starts 6 meters west of his house. He runs toward his house with a constant speed of 10 meters per second and continues running by his house for 5 seconds then stops.

[pic]

4) A woman leaves her house moving north at a constant velocity of 3 meters per second. After 5 seconds she gradually slows down until at 15 seconds she has stopped.

[pic]

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1.1.3 Acceleration

Definitions

• _____________________________: change in velocity occurring over time

– Measured in _______________________________

– ___________________

Direction of Velocity vs. Acceleration

The direction of velocity and acceleration do not have to be the same!!!

[pic]

Equations

Example

• A car starts with a velocity of -5 meters per second. At the end of 15 seconds, Car A has a velocity of +25 meters per second.

– What is the car’s acceleration?

– If a second car has the same acceleration and begins from rest, what is its speed after 4.0 seconds?

PRACTICE

1) A cheetah starts from rest and accelerates after a gazelle at a rate of 6.5 meters per second2 for 3.0 seconds. Calculate the cheetah’s speed at the end of these 3.0 seconds.

2) A truck moving at 7.0 meters per second accelerates to a speed of 15.0 meters per second within a time of 1.5 seconds. Calculate the truck’s rate of acceleration.

3) A driver moving with a speed of 15 meters per second sees a “Bridge Out” sign on the road ahead and begins to apply his brakes. His brakes decelerate the car at a rate of 2.0 meters per second2. How much time will it take for the car to come to a stop?

4) A skater begins moving down a ramp with some unknown speed. She accelerates down the ramp at a rate of 1.4 meters per second2. It takes the skater 3.2 seconds to reach the bottom of the ramp, at which time she has a speed of 6.5 meters per second. Determine the skater’s starting speed.

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GRAPHING PRACTICE

1) An object moving with a constant velocity of +5 meters per second that begins at its point of origin.

2) An object moving with a constant velocity of -2 meters per second that begins at a point +2 meters from the origin.

3) An object starting position at its point of origin and a starting speed of 0 that accelerates at +1 meter per second2.

1.1.4 Kinematics Equations

New Equations

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Example #1

• A train starts from rest and leaves Greenburg station and travels for 500 meters with an acceleration of 1.20 meters per second2.

– What is the train’s final speed?

– How long does it take the train to reach this speed?

Example #2

• A driver traveling in his car at 85 miles per hour sees a police car hiding in the trees 2 miles ahead. He applies his brakes and decelerates at a rate of 500 miles per hour2.

– If he is in a 55 mile per hour speed zone, will he get a ticket?

– What would his acceleration need to be in order to avoid the ticket?

PRACTICE

1) A runner begins from rest and accelerates at a rate of 0.22 meters per second2 for 15 seconds. Calculate the distance traveled by the runner while he accelerates.

2) A block sliding across the floor slows at a rate of -1.5 meters per second2. It takes the block 2.0 seconds to stop over a distance of 3.0 meters. Calculate its initial speed.

3) A 747 requires a takeoff speed of 270 meters per second if it is to get off the ground. Calculate the acceleration required for the 747 that begins from rest to takeoff on a 1000 meter long runway.

4) The scaled diagram below shows the scene of an accident that is being investigated by a CSI detective. Manufacturer data for the model of car that hit the pedestrian shows that typical deceleration for this car in a skid is about -6.5 meters per second2. The goal of this investigation is to find out whether or not the driver of the car was speeding when he slammed on his brakes and began to skid. It will also be important to find out how fast the car was moving when it hit the pedestrian.

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d. Determine the driver’s speed when he began to skid.

e. Calculate the speed at which the car hit the pedestrian.

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1.1.5 Freefall

Falling Down

- As objects fall they __________________________ (gain speed)

- They do this at a __________________________ no matter what the mass of the falling object is.

- We use “g” as a shorthand for ACCELERATION DUE TO GRAVITY

- On Earth the value for ‘g’ is ____________________.

- Other examples

o Moon: -1.62 m/s2

o Jupiter: -26 m/s2

o Mars: -3.75 m/s2

Using “g” to find changes in speed and distance

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Throwing Downward

An object is thrown downward from the top of a 175 meter building with an initial speed of 10 m/s.

[pic]

Throwing Upward

A cannon fires a shot directly upward with an initial velocity of 50 m/s.

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PRACTICE

1) Joe is standing on a bridge that spans a canyon drops a rock into the open space below him. Calculate the rock’s speed after it has fallen for 5 seconds.

2) Tina throws a softball straight up with an initial speed of 25 meters per second. Calculate the maximum height that the ball will reach. (There are a few ways to do this)

Circle the graph which best represents the velocity of the ball as it goes to its maximum height and then back to Tina’s hand.

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3) Captain Barker is an astronaut exploring Mars. He notices that when he drops a wrench from a height of 2 meters, it takes 1.03 seconds for it to reach the surface of the planet. Calculate the acceleration due to gravity on Mars.

[pic]

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-----------------------

• _______________________ - the complete length of a path traveled by a moving object.

• is a ___________________

• _______________________ - the length of the straight-line path of a moving object from its origin to its final position

• is a ___________________

A bird flies 5 meters north, then 7 meters south

Dist = _________

Disp= _________

A cat runs 8 meters west.

Dist = _________

Disp = _________

A ball rolls 5 meters north.

Dist = _________

Disp = _________

East bank

West bank

C

1 cm = 20 m

A

B

Review Questions

1.1.1a – Understand and explain the difference between distance and displacement.

- Two men leave the same house at the same time. Bill walks two blocks north, then four blocks south. Joe walks three blocks east, then one block north.

o Sketch a diagram of the their paths.

o Which man travels the greater distance?

o Which man finishes his trip with the greater displacement?

[Sketch; Bill; Joe]

- Explain how it is possible for a person to travel a great distance, and yet have a final displacement of zero.

[Explain]

- Two people travel from New York City to Boston. One person travels by plane, the other by car. Which person probably travels the greater distance during this trip? Compare the displacements of the two people assuming that they both start at JFK Airport and finish at Logan Airport.

[driver has greater distance

same displacement]

1.1.1b – Interpret graphs of distance or displacement vs. time. Use the graphs to determine average speed; displacement; or distance traveled.

During what time interval(s) is…

… the object’s average speed greatest?

…the object’s speed zero?

How far did the object travel between

3 and 6 seconds?

[3-4s; 1-3s & 6-9s; 5m]

What was the object’s direction of

travel during the time interval 7-8 s?

How far did the object travel between

4 and 8 seconds?

What was the object’s average speed

during the time interval 1-2 s?

What was the object’s position at time

t = 4 s?

What was the object’s final position?

[left; 6m; 2m/s;4m; 0m]

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B

A

30 m

20 m

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0

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Review Questions

1.1.2a – Understand and explain the difference between speed and velocity. Use equations to determine average speed and/or velocity.

- Two cars leave the same house at the same time. During the same interval of time: Car A travels three blocks east then three blocks west; while the Car B travels four blocks west, then one block north.

o Which car had the greater average speed? [A]

o Which car had the greater average velocity? [B]

- A cat moves 15 meters east in 5 seconds. What was its average speed?

[3 m/s]

- A fish swims 20 meters south in 10 seconds. What was its average velocity?

[-2 m/s]

- A jogger takes a 3.0 mile run. The first mile takes him 10 minutes to complete, the second mile takes him 11 minutes to complete, and the third takes him 9.5 minutes to complete. Determine his average speed.

[9.8 x 10-2 mi/min]

- An object begins with a speed of 6.0 meters per second and speeds up to 10 meters per second in 4.0 seconds. What is the object’s average speed during these 4.0 seconds?

[8.0 m/s]

- A boy walks 10 meters in 20 seconds and then 30 meters in 40 seconds. What is his average speed?

[0.67 m/s]

1.1.2b – Interpret graphs of speed or velocity vs. time. Use the graphs to determine average speed; acceleration; displacement; or distance traveled.

During what time interval(s) was…

…the object’s speed greatest?

…the object’s acceleration greatest?

…the object not moving?

How far did the object travel between

0 and 4 seconds?

[5-7s; 7-8s; 0s and 8-10s; 7m]

What was the object’s speed at time

t = 3 seconds?

During what time interval was the

object moving to the left?

What was the object’s average speed

during the time interval 1 to 2 seconds?

What was the object’s acceleration

during the interval 3 to 4 seconds?

Name one interval during which speed

was constant.

[2 m/s; 4-8s; 1.5 m/s; -2m/s2; 2-3s & 5-7s]

-

+ v

+ a

The car will…

- v

- a

The car will…

+ v

- a

The car will…

- v

+ a

-

The car will…

Acceleration is the rate of change in velocity

The slope of a velocity time graph tells us what the acceleration is doing.

Avoid this formula – except when interpreting graphs…

To know an object’s final velocity we need to know:

- its starting or initial velocity

- its rate of acceleration (if any)

- the amount of time that it accelerated for

Review Questions

1.1.3 – Explain the difference between velocity and acceleration. Use equations to determine acceleration; starting velocity; ending velocity; or time. Understand what the terms “from rest”; “comes to a stop”; “comes to rest” mean in terms of kinematics.

- An object begins from rest and reaches a velocity of +10 meters per second while accelerating for 4.0 seconds. Determine the acceleration of the object.

[2.5 m/s2]

- A car moving at 2.0 meters per second speeds up at a rate of 5.0 meters per second2 for 4.0 seconds. What is the final speed of the car?

[22 m/s]

- An object begins with a velocity of 20 meters per second west and comes to a stop within 4.0 seconds. Determine the direction and magnitude of the acceleration that acted upon this object.

[5 m/s2 EAST]

- An object with a velocity of 6.0 meters per second north accelerates south at a rate of 5.0 meters per second2 for 3.0 seconds. What is the final speed of the object? In which direction is the object moving after 3.0 seconds?

[9.0 m/s SOUTH]

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a

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d

t

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a

t

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t

a

“Distance or Displacement Equation”

“Timeless or Shortcut Equation”

1 cm = 5 m

Car came to rest here

Site where pedestrian was hit.

Car began to skid here

Review Questions

1.1.4 – Use equations to determine distance; displacement; speed; velocity; acceleration; or time.

- How far does an object travel if it starts from rest and accelerates at a rate of 2.0 meters per second2 for 6.5 seconds?

[42.25 m]

- What is the final speed of a cart that begins with a velocity of +3.0 meters per second if it accelerates at a rate of +1.5 meters per second2 while traveling +25 meters?

[84 m]

- At what rate does an object accelerate if it changes its speed from 3.0 meters per second to 5.0 meters per second while traveling a distance of 4.0 meter?

[2 m/s2]

- How long does it take for an object to reach a velocity of 2.0 meters per second north if it begins with a velocity of 8.0 meters per second south and accelerates north at a rate of 4.0 meters per second2?

[2.5 s]

t = 0 s

vi = 0 m/s d = 0 m

t = 1 s

v = _____________ d = _____________

t = 2 s

v = _____________ d = _____________

t = 3 s

v = _____________ d = _____________

An object falls from rest. What is its velocity after 1 second? Two seconds? Three Seconds?

An object falls from rest. How far has it fallen after 1 second? Two seconds? Three Seconds?

What acceleration does it experience?

____________

What is the object’s initial velocity?

___________

What is the object’s velocity as it hits the ground?

• vi = -10 m/s

• d = -175 m

• a = -9.81 m/s2

• vf = ?

How long does it take the object to hit the ground?

• t = ?

What is the object’s velocity when it reaches the top of its flight?

How long does it take the cannonball to reach the top of its flight?

What acceleration does the cannonball experience?

_______________

What is the cannonball’s initial velocity?

_______________

t = 1 s

t = 2 s

What is the maximum height of the cannonball?

t

v

t

v

t

v

t

v

Review Questions

1.1.5 – Explain the behavior of objects in freefall (in the absence of air resistance.). Understand the terms “dropped” and “maximum height”.

- What speed will a ball reach if it falls from rest for 3.0 seconds?

[29.43 m/s]

- How far will a dropped object have fallen after 2.0 seconds?

-

[19.62 m]

- What maximum height will an object reach if it thrown directly upward with a speed of 15 meters per second? How long will it take to reach this height?

[11.5 m]

- What is the height of an object that is thrown upward with a speed of 20 meters per second 1.5 seconds after it is thrown?

-

[18.96 m]

- An object is thrown directly upward with a speed of 25 meters per second. How long will it take to come back to the position from which it was released?

-

[5.1 s]

- How long does it take for an object that is dropped from the top of a 45 meter high building to reach the ground?

[3.0 s]

Review Questions

1.1.Xa – Relate graphs of: distance/displacement vs. time; speed/velocity vs. time; and acceleration vs. time.

Which sets of graphs could describe the motion of the same object?

[B; D; E; F; G; H; J]

t

v

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A

B

C

D

E

F

t

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t

a

t

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a

t

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G

H

I

J

Which graph would best represent…

a) speed vs. time for an unmoving object

b) displacement vs. time for an object moving with a constant speed

c) speed vs. time for a decelerating object

d) distance vs. time for an object with increasing speed

e) velocity vs. time for an object with a constantly increasing displacement

f) acceleration vs. time for an object that is moving at a constant speed

g) displacement vs. time for an object that is moving toward its point of origin

h) acceleration vs. time for an object that is increasing its speed at a constant rate

[a = B; b = C; c = D; d = E; e = A; f =B; g = D; h = A]

1.1.Xb – Give examples of vectors and scalars; recognize the difference between them.

- What two parts does a vector have? Which of these parts comprises a scalar?

[magnitude & direction;

magnitude only]

- Which of the following are vectors? Which are scalars?

a) distance (d) velocity

b) displacement (e) acceleration

c) speed (f) time

[vectors = b, d, e;

scalars = a, c, f]

F

E

D

C

B

A

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