D Change of Distance S = Step 1: Variables Step 3: Put in ... equal change of distance (distanced traveled) divided by ... the distance traveled and the elapsed time. ... A slower

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:1

Speed

Speed Speed is how fast something is moving. Precisely, it is how far an object travels in a certain amount of time. The standard metric units are meters per second (m/s), but any units of distance divided by time will work (like miles per hour [mph] or cm per sec [cps], etc).

S =

Speed equal change of distance (distanced traveled) divided by change of time.

Change of Distance (in meters)

Change of Time (in seconds)

Speed (in meter/sec)

!D

!T

Where !D = Dfinal ! Dinitial

Ex. A plane flies 200 meters in 5 sec. Calculate its speed.

Step 1: Variables S = ________ !D = 200 m !T = 5 sec Step 2: Formula

Step 3: Put in numbers and solve Step 4: Check units

S = 40 m/sec D

ST

!=

!

200

5

40

DS

T

S

!= =

!

=

Speed is proportional to distance: A faster object goes farther, in the same amount of time.

Speed is indirectly proportional to time: A faster object travels the same distance in less time.

Each dot represents an object’s position at regular time intervals (time is constant).

Measuring Speed Initial Position Final Position 25 m

Distance Traveled

0:05.0 Elapsed Time

5 sec 0:00.0

To measure speed you must measure the distance traveled and the elapsed time.

Measure distance in meters using a meter stick or measuring tape.

Measure time with a stopwatch or with photogates.

Photogates (which start and stop when an object breaks beams of light) are a very accurate and precise method of measuring time.

2 5 m5 m /s

5 sec

DS

T

!= = =

!

100m in 10sec

200m in 10sec

1

10010m/s

10

DS

T

!= = =

!

2

20020m/s

10

DS

T

!= = =

!

Doubling the distance, doubles the speed.

200m in 20sec

200m in 10sec 2

20020m/s

10

DS

T

!= = =

!

1

20010m/s

20

DS

T

!= = =

!

Doubling the time, halves the speed.

Constant Speed

A slower object can travel the same distance as a faster object, it just takes more time. A fast object travels the same distance faster.

If an object moves at constant speed, it travels the same amount of distance each second. Notice that there is equal space between each dot.

Why we use change of distance: A tree 4 m away for 2 sec has a speed of zero — it hasn’t moved. That’s why we have to use !D (change of distance) instead of distance (D). An object has to be moving to have speed.

Physics Explains Mathematics: If !T = 0 (in S = !D/!T), then an object is in two places at once, which is impossible. This is why dividing by zero is undefined: it makes no physical sense!

Fast object

Slow object

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:1

1. Speed

2. Distance Traveled

3. Elapsed Time

4. !

5. Constant Speed

A. How far an object moves between two positions.

B. When an object covers equal amounts of time each second.

C. The rate of how fast an object travels a particular distance.

D. How many seconds it takes for an event to occur.

E. Delta: means “change of”.

True or false (and why): “A fast car goes farther.” Can a slow object travel as far as a fast object? Explain. Why do we have to use change of distance (!D) instead of just distance (D)?

A bike moves 50 m in 10 seconds. Calculate the speed of the bike.

Step 1: Variables: S = !D = !T = Step 2: Formula:

Step 3: Plug in numbers and solve: Step 4: Give answer with units:

1. Slow speed

2. Fast speed

3. Photogate

4. Directly Proportional

5. Indirectly Proportional

A. An object that travels a long distance quickly.

B. Can travel a long distance, but requires a lot of time.

C. Uses a beam of light to start and stop a timer.

D. One quantity increases as another quantity increases.

E. One quantity decreases as another quantity increases.

_____ 5 mm/sec

_____ 10 inches

_____ 50 m/s2

____ 20 meters/sec

____ 228 meters

____ 8 minutes

____ 15 ft/min

____ 78 sec

____ 6 Newtons

Mark these as Speed, Distance, Time, or Other

A car travels 60 m/s for 10 secs. Calculate how far it traveled.

Step 1: __________ Step 2: __________

Step 3: ______________________ Step 4: ______________________

On holiday, a family travels from Meyerville (10 miles away) to Sprytown (70 miles away), in 3 hours. Find their speed.

Step 1: __________ Step 2: __________

Step 3: ______________________ Step 4: ______________________

A car travels 200 miles in 4 hours. Calculate the car’s speed.

Step 1: Variables: S = !D = !T = Step 2: Formula:

Step 3: Plug in numbers and solve: Step 4: Give answer with units:

____ Distance is constant and time increases.

____ Time is constant and distance decreases.

____ Time is constant and distance increases.

____ Distance is constant and time decreases.

Will Speed Increase or Decrease?

1. Is the above motion at constant speed? 2. Why or why not?

3. Each dot = 1 sec. How long did it take to go 15 m? 4. Calculate the object’s speed.

5. How would the dots change if it were moving faster?

start

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:2 Velocity and Acceleration

Example: A person walks 4 m/s—speed (no direction).

Speed vs. Velocity Velocity is speed with direction.

20 m/s north

20 m/s west

Same speed; different velocities because they

have different directions.

Scalars vs. Vectors

Remember: Speed is a Scalar; Velocity is a Vector.

Vectors require direction; Scalars only need magnitude (how big).

Vectors require magnitude (how much) and direction, often vectors can cancel each other out (not acceleration, though).

12 m/s west Magnitude Direction

Speed: 12 m/s. Velocity: 12 m/s west. Velocity changes when direction changes.

Ex. A plane starts at rest and ends up going 200 m/s in 10 secs. Calculate its acceleration.

Step 1: Variables Vi = 0 m/s (at rest) Vf = 200 m/s T = 10 sec a = _________

Step 2: Formula

V

aT

!=

!

Step 3: Put in numbers and solve Step 4: Add units

a = 20 m/s2

200 0

10

20020

10

f iV VVa

T T

a

"! "= = =

! !

= =

Acceleration

Acceleration is how fast you change velocity OR how much the velocity changed in a certain amount of time. An object accelerates when it changes speed OR changes direction!

a =

Acceleration equal change of velocity divided by change of time.

Change of Velocity (in meters/sec)

Change of Time (in seconds)

Acceleration (in m/s2)

!V !T

, so, final initial

final initial

V VV V V a

T

"! = " =

!

Ex. A race car starts at 40 m/s slows to 10 m/s in 5 seconds. Calculate the car’s acceleration.

Step 1: Variables Vi = 40 m/s Vf = 10 m/s T = 5 sec a = _________

Step 2: Formula

V

aT

!=

!

Step 3: Put in numbers and solve Step 4: Add units

a = –6 m/s2

10 40

5

306

5

f iV VVa

T T

a

"! "= = =

! !

"= = "

Neg. means slowing down

Negative acceleration

means an object is slowing down OR speeding up in the negative

direction. Slowing down is also called

“deceleration”.

Finding !V.

! always = final – initial. !V = Vfinal – Vinitial OR

Final velocity – Initial velocity.

If !V is positive the object is speeding up.

If !V is negative the object is slowing down (see below).

Distance and Acceleration

Pos. means speeding

up

Measuring Acceleration

To measure an object’s acceleration you need to

measure the object’s velocity before and after

the acceleration.

If the object starts at rest you know that Vi = 0m/s.

If the object stops you know that Vf = 0m/s.

Points are equal distance, so velocity is constant. Since the velocity is constant, the initial and final velocity

are equal and the acceleration equals zero.

The distance between the points is increasing, so velocity is increasing. The object is accelerating: traveling faster each second and covering more distance every second.

An object that is accelerating will travel farther each second.

4 m

1 sec

4 m /s

i

in it ia l

DV

T

V

!= =

!

=

8 m

1 se c

8 m /s

f

fin a l

DV

T

V

!= =

!

=

2

8 4

2

42m/s

2

f i

initial

V Va

T

V

" "= =

!

= =

Constant Speed—Equal Distance Positive Acceleration—Increasing Distance

Accelerates for 2 seconds So !T = 2 sec

4 m in 1 sec

Measure Vf (Final Velocity)

8 m in 1 sec

Measure Vi (Initial Velocity)

Measure !T (Time it took to Accelerate)

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:2

Mass, Time, Distance, Velocity, or Acceleration?

___ 2 hrs

___ 3 m/s

___ 6 mph/sec

____5 sec

____9 mph

____12 m

____8 kg

____4 m/s2

____1 in

___A bike goes 25 m/s toward main street.

___A person walks 4 mph.

___A plane flies 200 m/s.

___A bird flies 100 mph due south.

Speed (S) or Velocity (V)

___ 40 mph toward Dallas.

___ 3 m/s2 to the left.

___ 10 meters up the hill.

___ 12 meter per sec2.

___ Direction matters.

___ No direction is needed

Scalar (S) or Vector (V)

A dragster’s top acceleration is 60 m/s2. If it starts from rest at the starting line, how fast will it be going after 3 seconds?

Variables: Formula:

Solve:

A person starts running from 2 m/s to 6 m/s in 2 seconds. Calculate the person’s acceleration.

Variables: Formula:

Solve:

A car travels 30 m in 5 seconds. After accelerating for 3 seconds, it travels 20 m in 2 seconds. Calculate the car’s acceleration.

1) Find Vi. 2) Find Vf. 3) Calculate a.

A plane stops from 250 mph in 25 seconds. Calculate the planes acceleration.

Variables: Formula:

Solve:

10 m/s

10 m/s

Accelerating? Yes, No, or Maybe?

___ At constant velocity.

___ Going 5 m/s then going 3 m/s.

___ A car going around a corner. (see graphic at right).

___ At constant speed.

___ Stopping.

___ A car at rest.

Object A accelerates at 10 m/s2; Object B accelerates at 5 m/s2. ___ Which one will go faster?

___ Which one will take more time to reach a high speed?

___ If they start at rest, which one will reach 40 m/s first?

___ Which one goes farther (longer distance)?

___ Which one will be 100m away sooner?

Object A

Object B

Object C

Choose which of the above applies to the following

____ Constant speed. ____ Positive acceleration. ____ At constant velocity. ____ Accelerating. ____ Decelerating. ____ Acceleration = 0.

____ Distance increases ____ Starts at rest. ____ Is stopping. ____ Constant direction. ____ Negative acceleration. ____ Vi = Vf

Object D

Give what you know for the following: (Vi, Vf, or a)

An object at constant velocity.

An object that is stopping.

An object that accelerates from rest.

An object at rest.

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:3

Graphing Linear Motion

Conventions: X-axis (horizontal): Independent or manipulated variable. Y-axis (vertical): Dependent or responsive variable.

Meaning of Slope Changes The slope of a position vs. time graph is speed. The slope of a velocity vs. time graph is acceleration. Yet for some graph, the slope has no physical meaning.

Position vs. Time Graphs

Graphing Variables

A Position vs. Time graph shows where an object is at a particular time. The slope of a position vs. time graph shows the speed of an object. A steeper line shows faster speed. A downward line means negative speed (moving left or coming back).

A steeper line = a faster speed.

306m/s

5LineA

DS

T

!= = =

!

303m/s

10LineB

DS

T

!= = =

!

Object B travels 30 m in 10 seconds. Line B shows slow positive speed.

Position vs. Time

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (sec)

Po

sit

ion

(m

)

Line A fa

st sp

eed

slow sp

eed Line B

negative speed Line D

Starting position (t = 0)

no speed Line C

Object C stays 15 m away. Line C shows a speed of zero.

00m/s

10LineC

DS

T

!= = =

!

Object D travels –20 m in 10 seconds. Line D shows slow negative speed.

202m/s

10LineD

DS

T

! "= = = "

!

Object A travels 30 m in 5 seconds. Line A shows fast positive speed.

To figure out what the slope of a graph means:

divide the y-axis units by the x-axis units to find the

units for the slope.

Scientists have rules for choosing which variable is graphed on which axis. This allows scientists to understand how an experiment was conducted just by reading the graph.

Independent vs. Dependent

The independent vari-able is not affected by the changing depend-ent variable. The de-

pendent variable changes as the inde-

pendent variable

Manipulated vs. Responsive

Sometimes it is hard to determine which is the

independent variable. In these cases, the variable

that you are manipulating (varying) will graphed on

the x-axis.

Velocity vs. Time

Dep

end

ent

vari

ab

le

Vel

oci

ty (

in m

/s)

Time (in sec) Independent variable

Acceleration vs. Force

Res

po

nsi

ve v

ari

ab

le

Acc

eler

atio

n (

in m

/s2)

Force (in N) Manipulated variable

The above object’s acceleration changes (responds) as the force is

changed (manipulated).

This graph shows the change of acceleration over time which is undefined.

Acceleration vs. Time

Acc

eler

atio

n

(in

m/s

2)

Time (in sec)

23m/s

m/s ?s

rise ySlope

run x

!= = = = =

!

Velocity vs. Time

Vel

oci

ty (

in m

/s)

Time (in sec)

This graph shows the change of velocity over time which is acceleration.

2m/sm/s acceleration

s

rise ySlope

run x

!= = = = =

!

Slope = !acceleration

Time (as in “a particular moment in time”) is always an independent variable (x-axis)

because nothing stops time. Time does not change with

speed; speed changes over time.

Duration (how long it takes) can be dependent (y-axis). Ex. The period of a spring (how long it takes to move back and forth) changes as more mass is added. Mass is inde-pendent, not period of time.

The slope of this graph

means nothing.

The manipulated variable is the one you are changing in your ex-periment and is often the experi-mental variable.

Meaning of Slope

units of y-axis

units of x-axis

rise

run=

=

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:3

When was the object moving at 150 m/s? ______________________

How fast is the object going after 10 seconds? __________________

What was the initial velocity of the object? _____________________

How much speed does it gain in the first 5 seconds? ______________

Find the slope of the graph (must show work) ___________________

What does the slope you just found stand for? __________________

1. Linear

2. Responsive variable

3. Independent variable

4. Dependent variable

5. Slope

6. Manipulated variable

A. Vertical axis (y) variable.

B. The variable you change.

C. Any straight line graph.

D. Measure of how steep a line is.

E. The variable on the horizontal axis (x-axis).

F. What changes because you change something.

Position vs. Time

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Time (sec)

Po

sit

ion

(m

)

What does the slope of this line show? ________________________

How much time does it take Object A to travel 100m? ___________

How much time does it take Object B to travel 100m? ___________

Which Object (A or B) has the faster velocity? _________________

Object C starts where? ________ Object C ends where? _________

Which line shows negative speed? ___________________________

Which line shows positive speed? ___________________________

Which line shows an object at rest? __________________________

What is Object D’s initial position? __________________________

Which is the independent variable? ___________________________

Which is the dependent variable? _____________________________

Where was the object at 4 seconds? ___________________________

Where did the object begin? _________________________________

Find the slope of the graph (must show work)

What does the slope you just found stand for? ___________________

The slope of this graph means:

Which segment shows: Increasing velocity:

Constant velocity:

Positive acceleration:

Negative acceleration:

Speeding up:

Slowing down:

Position vs. Time

Time

Po

siti

on A

B C D

Which segments shows:

At rest:

Fast speed:

Slow speed:

Going backwards:

Going forward:

Negative speed:

Speed equals zero:

Position vs. Time

0

2

4

68

10

12

14

16

18

0 1 2 3 4 5 6Time (sec)

Po

siti

on (

m)

Velocity vs. Time

Time

Vel

oci

ty

A B

C D

Circle the Independent Variable A. Time or Acceleration B. Velocity or Time C. Time or Position

Circle the Manipulated Variable for these Graphs

A. Force on an object or Acceleration of the object? B. Period of a Spring or Mass hung from the spring? C. Number of batteries or Brightness of a bulb?

Velocity vs. Time

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10 11

Time (secs)

Ve

loc

ity

(m

/s)

A B

C

D

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:5 Linear Motion Review

mv = m times v

F/a = F _______ a

T2 + T1 = T2 _______ T1

mv = m _______ v

!D/!T = !D _______ !T

Match the variables with the quantities.

1. a = _____________

2. S or v = _____________

3. D = _____________

4. F = _____________

5. T = _____________

sec

m/sec

43 m/s2

45 meters

22 newtons

Equation: S = !D/!T; solve for !D. If !v = v2 – v1, solve for v2:

If p = mv, solve for m. If a = !V/!T, solve for !T:

For the following problems, show all work and steps. A plane stops from 300 mph in 15 seconds. Calculate the planes acceleration. A bike going 3 m/s ends up going 9 m/s after 2 seconds. Calculate the bike’s acceleration.

A car travels 35 m in 5 secs. Calculate its speed.

Variables: Formula:

Solution:

A bike goes 12 m/s for 6 seconds. Calculate how far the bike traveled.

Variables: Formula:

Solution:

A

B

C

D

E Choose which of the above

object’s motion applies to the following (can be more than one):

______Vi = 0 ______Decelerating ______Constant speed ______Is stopping ______Positive acceleration ______At constant velocity ______Vf = 0

_____ Accelerating _____ Acceleration = 0 _____ Distance is increasing _____ Starts at rest _____ Constant direction _____ Negative acceleration _____ Vi = Vf

For object B above: A) If there is 1 second between each dot, when did the object reach 12 m? B) Find the speed of object B.

What do you need to know in order to find an object’s speed? What does ! mean (and give the formula)?

Which has the faster speed? Car A or Car B? Both go the same distance, but Car B gets there sooner. In the same amount of time, Car A goes farther. TA = TB, but DA < DB.

Car 1 is going 20 m/s. Car 2 is going 30 m/s. Which one travels 100 m first? Which one can travel a greater distance? Which one travels farther in more time?

An object has a velocity of 5 m/s and starts 0 m away from you. A) How far does it travel each second? B) Where is it after 1 second? C) Where is it after 2 seconds? D) Where is it after 5 seconds? E) How far does it travel between seconds 7 and 8?

Speed (S) or Velocity (V)

____ A car travels 10 m/s left.

____ A bird flies 20 m/s.

____ A bike goes 10 m/s toward town.

____ 10 m/s.

____ 60 mph toward Austin.

____ Direction matters.

Scalar (S) or Vector (V)

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Unit 6:5

Position vs. Time

Time

Po

siti

on

Which graph segments fit the following:

At rest:

Fast speed:

Slow speed:

Going backwards:

Going forward:

Speed vs. Time

Time

Sp

eed

Which graph segments fit the following:

Constant speed:

Negative acceleration:

Positive Acceleration:

Slowing down:

Acceleration = 0:

For the following problems, show all work and steps. A 4 kg object is moving 6 m/s to the left. Calculate momentum. A 10 kg object has 58 kgm/s of momentum. Find its velocity. Find the net momentum of the two objects at the right.

Which is the independent variable? __________________________

Which is the dependent variable?_____________________________

Where was the object at 4 seconds? __________________________

Where did the object start? _________________________________

When did the object reach 8 meters? __________________________

Find the slope of the graph (show work) What does the slope you just found stand for? __________________

Position vs. Time

02468

101214161820

0 1 2 3 4 5 6

Time (sec)

Line A

An object accelerates at 10 m/s2. Answer the following: A) If it starts at rest, how fast is it going after 1 second? B) After 2 seconds, how fast is it going? C) If it starts at 5 m/s, how fast would it be going after 1 second?

Fast car

Fast truck

Fast plane

Fast hammer

A mountain

Number these from most (1) to least (5) momentum.

If two objects have a net momentum of 45 kgm/s before they collide, how much momentum will they have after they collide? An astronaut is by herself in space. All she has is a box of tools. How can she get to her ship that is to her left? How is it possible that two moving objects can collide and stop moving? A 200 kg cannon shoots a 2 kg cannonball. If the ball ends up going 300 m/s to the right: A) If they are both at rest beforehand, what is "pbefore? B) What is "pafter? C) Is the ball’s final p positive or negative (pball A)? D) Is the cannon’s final p positive or negative (pcannon A)? E) Find the velocity of the cannon afterwards vcannon A)?

10 kg

4 m/s

8 kg

3 m/s

Po

siti

on (

m)

A

B

C

D

D

C

A

B

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.

Name: _____________________

Period: _____________________

cstephenmurray.com Copyright © 2008, C. Stephen Murray Legal copying of this worksheet requires written permission.