Unit B01 – Motion in One Dimension [UNAUTHORIZED COPYING OR USE OF ANY PART OF ANY ONE OF THESE SLIDES IS ILLEGAL.]

Unit B01 – Motion in One Dimension

[UNAUTHORIZED COPYING OR USE OF ANY PART OF ANY ONE OF THESE SLIDES IS ILLEGAL.]

Section 2-1: Reference Frames and Displacement

Explain what “frame of reference” is in terms a junior-high student could understand.

A frame of reference is like a point of view. Specifically, it is the point of view of an observer making measurements and describing what is going on.


Explain what “frame of reference” is in terms a junior-high student could understand.

Example: Adam (red) says the lightning bolt struck 2 mi north and 2 mi west of him. Betty (blue) says the lightning struck 4 mi south and 3 mi west. Both give a different position for the lightning bolt. Who is right?

Both observed the same thing, but made different measurements based on their different frames of reference.


Explain the difference between “distance” and “displacement”.

Distance – The length of the path an object travels.

Displacement – Distance straight from start to end, with direction specified.


Explain the difference between “distance” and “displacement”.

• What is Adam’s (red) distance from the lightning strike?

• What is Adam’s displacement from the lightning strike?

• What is Betty’s (blue) distance from the lightning strike?

• What is Betty’s displacement from the lightning strike?

• What is the lightning’s displacement from Adam?

• What is Adam’s distance from Betty?


Explain what a vector is.

A vector is any measured quantity that has both a magnitude (amount) and a direction.

Explain what a scalar is.

A scalar is any measured quantity that has only a magnitude (amount) but no meaningful direction.


What letter is the symbol for displacement used in mathematical equations?

xDistance and displacement are measured in what units?

Meters


Example: Consider the coordinate axes below. Each box represents 1 meter.

y

x

Fred’s position is Point A, which is at (x = 3, y = –3). Mark this point.

A



Fred walks 5 meters to the left, 5 meters up, and 5 meters to the right, arriving at point B. Mark point B and Fred’s path on the diagram.

y

x

A

B

A. (2, 2)B. (–2, 2)C. (–2, –3)D. (–3, –2)E. (2, 3)F. (3, 2)



What distance did Fred travel?y

x

A

B

15 m

What is the distance between points A and B?

5 m

What is Fred’s displacement from his original position?

5 m North

A. 5 metersB. 10 metersC. 15 meters

D. 5 meters NorthE. 10 meters WestF. 15 meters North

Section 2-2: Average Velocity

Give a word-equation for average speed.

elapsed time

traveleddistancespeed average

Give a word-equation for average velocity.

elapsed time

ntdisplacemenet velocityaverage


What is the difference between speed and velocity?

Speed is a scalar—it only has a magnitude (such as 60 mph).

Velocity is a vector—it has magnitude and direction (such as 60 mph north).


Which is a true statement?

(A) Average speed can be greater than average velocity.

(B) Average velocity can be greater than average speed.

A person travels 10 m east, then 5 m west in 10 seconds.

His displacement is 5 m (east), but the distance he traveled is 15 m.

The first one.

This makes his average velocity (5 m east)/(10 sec) = 0.5 m/s east.

His average speed is (15 m)/(10 sec) = 1.5 m/s.


The symbol for time is t. Time is measured in seconds.

The symbol for velocity is v. Velocity is measured in meters/second.

“Average” is indicated by putting a straight bar over the symbol v.

v“Change” is represented by the letter “delta” ()

“Change” is determined by subtracting:

(value) = (final value) – (initial value)



y

x

Frank’s position is Point C, which is at (x = 4, y = 4). Mark this point.

C



y

x

Frank walks down at a speed of 2 m/s for 4 seconds.

He then immediately walks left 3 m/s for 3 seconds.

He waits for 5 seconds, and then walks 1 m/s upward for 8 seconds and stops at point D.

Plot point C, point D, and Frank’s path.

CD

A. (4, –4)B. (–4, 4)C. (–5, –4)

D. (–5, 4)E. (5, –4)F. (–4, –4)



y

x

During this time, what distance did Frank travel?CD

25 mWhat is the distance between points C and D?

9 mWhat is Frank’s displacement from his original position?

9 m West

A. 8 metersB. 9 metersC. 25 meters

D. 8 meters NorthE. 9 meters WestF. 25 meters West



y

x

What is Frank’s average speed during this time?CD

(Avg. Speed) = (Distance)/(Time)

(Avg. Speed) = (25)/(20)

(Avg. Speed) = 1.25 m/s

A. 0.45 m/sB. 1.25 m/sC. 1.67 m/s

D. 0.45 m/s WestE. 1.25 m/s WestF. 1.67 m/s West



y

x

What is Frank’s average velocity during this time?CD

(Avg. Vel.) = (Displacement)/(Time)

(Avg. Velocity) = (9)/(20)

(Avg. Velocity) = 0.45 m/s West

A. 0.45 m/sB. 1.25 m/sC. 1.67 m/s

D. 0.45 m/s WestE. 1.25 m/s WestF. 1.67 m/s West


Originally, Frank waited for 5 seconds (below point D). If he wanted to make his average speed half as much, how long should he have waited? Explain your reasoning.

A. 5 secondsB. 10 secondsC. 15 secondsD. 20 secondsE. 25 secondsF. 30 seconds


Originally, Frank waited for 5 seconds (below point D). If he wanted to make his average speed half as much, how long should he have waited? Explain your reasoning.

elapsed time

traveleddistancespeed average

…double the denominator.In order to cut this in half…

The total time from before was 20 seconds.

To cut the original avg. speed in half, the total time needs to be 40 sec.

This adds 20 seconds to the original wait time of 5 seconds.

New wait time: 25 seconds.


Would this amount of wait-time have also halved his average velocity?

This was doubled……so this is also cut in half.

Answer: YES.

elapsed time

ntdisplacemenet velocityaverage

Section 2-3: Instantaneous Velocity

Explain what “instantaneous velocity” is in the simplest terms possible. Make sure you point out the difference between this and “average velocity”.

“Instantaneous velocity” is the rate at which an object moves at a particular point in time.

“Average velocity” is the rate at which an object moves over a particular interval of time.


What does a speedometer in a car measure?

Instantaneous Speed – It tells you your rate of motion right now, but does not give you any information about direction.

A. Average SpeedB. Average VelocityC. Instantaneous SpeedD. Instantaneous Velocity


What two instruments in a car can be used to measure the car’s average speed during a long car trip? Explain how to use these instruments to calculate average speed.

Odometer (mileage gauge) – Measures the distance of the car trip.

Clock– Measures the time of the car trip.

Take the total distance traveled and divide by the total time of the trip to get the average speed over a long car trip.

Section 2-4: Acceleration

Write a word-equation for acceleration.

elapsed time

yin velocit changeonaccelerati

The symbol for acceleration is a. The units are m/s2.


Explain where the “square” in the units for acceleration comes from.

elapsed time

yin velocit changeonaccelerati

These units are m/s

These units are s

Plug in the units of m/s and s to get:

s

m/sonaccelerati

2s

m

s

1

s

monaccelerati


Explain the difference between velocity and acceleration using the words “rate of change of”.

Velocity – Rate of change of an object’s position. (Or: the rate at which an object’s position changes.)

6 m/s means that the object gains 6 meters of position every second.

Acceleration – Rate of change of an object’s velocity. (Or: the rate at which an object’s velocity changes.)

6 m/s2 means that the object gains 6 m/s of speed every second.


Acceleration is a vector. Explain how to determine the direction of acceleration.

If an object is speeding up, then acceleration points in the same direction as the object’s velocity.

If an object is slowing down, then acceleration points in the direction opposite the object’s velocity.


Example: Each diagram shows a car’s position at one second intervals as it travels along a line in the same direction. The solid [white] car is the final position at time = 5 seconds. For each diagram, make a graph of position vs. time, and draw an arrow indicating the direction of velocity and an arrow for acceleration:

0 5 10 15 20 25

x

t0

5

10

15

20

25

1 2 3 4 5

Velocity Acceleration

(A) (B) (C) (D) (E) 0(F) 0



x

t0

5

10

15

20

25

1 2 3 4 5

0 5 10 15 20 25


(A) (B) (C) (D) (E) 0(F) 0



x

t0

5

10

15

20

25

1 2 3 4 5

0 5 10 15 20 25


(A) (B) (C) (D) (E) 0(F) 0



x

t0

5

10

15

20

25

1 2 3 4 5

0 5 10 15 20 25


(A) (B) (C) (D) (E) 0(F) 0

Section 2-5: Motion at Constant Acceleration

Write the equation for velocity as a function of time for constant acceleration.

Write the equation for average velocity under constant acceleration.

0vatv

t

xv

2

0vvv


Write the equation for position as a function of time for constant acceleration.

Write the equation that relates velocity, acceleration, and position (the “no time” equation).

002

21 xtvatx

200

2 2 vxxav


Identify every symbol that you used in the four equations above.

x = Position

x0 = Initial Position

v = Velocity

v0 = Initial Velocity

a = Acceleration

t = Time


Section 2-6: Solving Problems

Example: A runner finishes a 100-meter dash in 8 seconds. Determine his average speed.

x = 100 m, t = 8 sec

v = ?

(100) = v(8)

G:

U:

E: tvx

SS:

v = 12.5 m/s


Example: A car can go from zero to 30 m/s in 2 seconds. Determine the car’s average acceleration.

v0 = 0, v = 30 m/s, t = 2 sec

a = ?

(30) = a(2)

G:

U:

E: 0vatv

SS:

a = 15 m/s2


Example: Another car can go from zero to 30 m/s in 3 s. How far does the car go while it accelerates?

v0 = 0, v = 30 m/s, t = 3 sec

x = ?

x = (15)(3)

G:

U:

E:

SS:

x = 45 m

tvx Avg. speed = average of start & end speeds


Example: The radius of Earth’s orbit is 1.5 1011 m. How fast does Earth move as it orbits the Sun?

r = 1.5 1011 m

v = ?

2(1.5 1011) = v(31536000)

G:

U:

E:

SS:

v = 29886 m/s

vtx x = circumference = 2r

t = 1 year = 365 days = 31536000 seconds

Section 2-7: Falling Objects

What type of motion is free-fall?

Constant Acceleration

A. Constant PositionB. Constant VelocityC. Constant AccelerationD. Changing Acceleration


What does the symbol mean in the top paragraph on page 33 that states “d t2”? How is that different than the = symbol?

The symbols says “is proportional to”. It means that two symbols have the same ratio all the time.

“d t2” means that d/t2 = (the same number all the time).

When accelerating, d = ½at2, so d/t2 = ½a (the same number all the time).

If d = t2, then d/t2 = 1 all the time.

For a circle, A = r2. This means that A/r2 = (the same number all the time), so A r2


Explain Figure 2-17 on page 33. Why does the paper fall differently than the ball in one case, but not in the other?

The paper’s surface area contributes to the force of air resistance on it. When the paper is balled up, air resistance is greatly reduced and it falls at the same acceleration as the ball and everything else in free-fall.


What is the value of the acceleration of gravity on Earth’s surface? What symbol do we give this number?

The acceleration of gravity on earth is 9.8 m/s2. We give this value the special symbol g.

When a calculator is not available to us (or sometimes when it is just convenient), we use g = 10 m/s2.


Example: A person throws a ball upward near the edge of a building as shown. Ignore air resistance. The first white dot is the position of the ball when it leaves the thrower’s hand. Each white dot after that represents the ball’s position every second after the ball leaves the thrower’s hand. In the table below, the time represents how many seconds after the ball leaves the thrower’s hand.

Time(seconds)

Position x

(meters)

Speed(m/s)

Velocity(m/s)

Accel.(m/s2)

0

1

2

3

4

5

00–10–20–30

102020

10

102030

01520150

–25

–10–10–10

–10–10–10


Fill in the boxes representing the ball’s velocity at each time.

When the ball is at ____________________, we know for sure its velocity is ________________.

Each second, the ball’s velocity ______________________________________________________.

The speed is almost the same as velocity, except __________________________________________________.

Fill in the ball’s acceleration at each time. How does the ball’s acceleration vary with time? ____________.

the highest heightzero

decreases by 10 m/s (has –10 m/s added to it every sec)

speed has no direction and is always positive

It doesn’t

Section 2-8: Graphical Analysis of Linear Motion

How can a curve be considered to have a slope?

We look at a point on the curve, and create a line tangent to the curve at that point. The slope of the tangent line is the same as the slope of the curve at that point.

y

x


How can velocity at a certain time be found by looking at a position vs. time graph?

Velocity is the slope of a position vs. time graph.

x

t

Going slow in the positive direction

Stop for an instantGoing fast in the positive direction

Going fast in the negative direction


How can the change in position between two times be found from a velocity vs. time graph?

On a velocity vs. time graph, the area between the graph and the horizontal axis gives the change in position of an object.

v (m/s)

t (m/s)012345

–1–2–3–4–5

1 2 3 4 5 6 7 8 9 10

This area (of 12) means the object went 12 m in these 5 sec.

This area of –6 means the object went 6 mto the left in these 3 seconds.


How can acceleration at a certain time be found by looking at a velocity vs. time graph?

Acceleration is the slope of a velocity vs. time graph.


How can the change in velocity between two times be found from an acceleration vs. time graph?

On an acceleration vs. time graph, the area between the graph and the horizontal axis gives the change in velocity of an object.


Example: Consider the following x vs. t graph.x (m)

t (sec)

5

0

–5

10 20

Estimate the velocity at:

t = 4 ________ t = 10 ________ t = 12 ________–2 m/s 0 m/s 1 m/s



t (sec)

5

0

–5

10 20

During what time intervals is the velocitypositive? ______________________________negative? ___________________________zero? ___________________________

10 to 21 sec0 to 8 sec8 to 10 sec



t (sec)

5

0

–5

10 20

During what time intervals is the accelerationpositive? ____________________________negative? ___________________________zero? ___________________________

4 to 8, 10 to 12, 14 to 16 sec0 to 4, 12 to 14, 19 to 21 sec8 to 10, 16 to 19 sec

Section 2-8: Graphical Analysis of Linear Motionx (m)

t (sec)

5

0

–5

10 20

Velocity

Negative Positive

Acc

eler

atio

n

Neg

ativ

eP

osit

ive



t (sec)

5

0

–5

10 20

What is the object’s average velocity between 0 and 10 seconds? _______________What is the object’s acceleration between 0 and 4 seconds? _______________

–0.8 m/s

–1 m/s


Example: Consider the following v vs. t graph.v (m/s)

t (sec)

5

0

–5

10 20

What is the acceleration at:t = 11 s? __________ t = 15 s? __________ –2 m/s2 0 m/s2



t (sec)

5

0

–5

10 20

How far, and in which direction, did the object travel:from t = 4 to t = 11 s? __________ from t = 13 to t = 18 s? __________

24 m right16 m left

Area = 24

Area = –16



t (sec)

5

0

–5

10 20

During what time intervals is the objectspeeding up? ______________________________slowing down? ______________________________

0 to 4, 11 to 13 seconds9 to 11, 16 to 18 seconds



t (sec)

5

0

–5

10 20

During what time intervals is the accelerationpositive? ______________________________negative? ____________________________zero? ____________________________

0 to 4, 16 to 18 seconds9 to 13 seconds5 to 9, 13 to 16 seconds



t (sec)

5

0

–5

10 20

If the object is at x = –5 m at t = 0 s, where is the object at t = 20 s? __________

Area = 8 Area = 20 Area = 4

Area = –4

Area = –12

Area = –4

Net Area = 12

+7 m

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = 1

Value = 1

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = 0

Value = 0

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = –2

Value = –2

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = 0

Value = 0

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = 2

Value = 2

v (m/s)

t (sec)

5

0

–5

10 20

a (m/s2)

t (sec)

1

0

–110 20

2

–2

Slope = 0

Value = 0

v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = 8

Value = 8Now x =3

v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = 20

Value = 20Now x = 23

v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = 4

Value = 4Now x = 27

v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = –4

Value = –4Now x = 23

v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = –12


v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = –4


v (m/s)

t (sec)

5

0

–5

10 20

x (m)

t (sec)

5

0

–510 20

10

15

20

25

Area = 0

Value = 0Still x = 7

Example: For each pair of graphs, use the given graph to make a sketch of the other graph. Dashed lines are for reference.

x

t

v

t

SLOPE

VALUE

SLOPE starts positive anddecreases to zero

SLOPE goes from zero intothe negatives.

VALUE starts positive anddecreases to zero

VALUE goes from zero intothe negatives.


v

t

a

t

SLOPE

VALUE

SLOPE starts negative andgoes to zero

SLOPE goes from zero intothe positives.SLOPE is zero.

VALUE starts negative andgoes to zero VALUE is zero.

VALUE goes from zero intothe positives.


a

t

v

t

VALUE

SLOPE

VALUE starts at zero and

goes positive

VALUE is a constant positive

VALUE goes from

positive to zero

SLOPE starts at zero and

goes positive

SLOPE is a constant positive

SLOPE goes from

positive to zero


v

t

x

t

VALUE

SLOPE

VALUE starts

negative and goes to zero

VALUE goes from zero to a

maximum positive

VALUE goes from

zero to negative

VALUE goes from maximum positive to zero

SLOPE starts

negative and goes to zero

SLOPE goes from zero to a

maximum positive

SLOPE goes from maximum positive to zero

SLOPE goes from

zero to negative

Documents

Unit B01 – Motion in One Dimension [UNAUTHORIZED COPYING OR USE OF ANY PART OF ANY ONE OF THESE SLIDES IS ILLEGAL.]