Chapter 2: Motion in one dimension

CHAPTER 2: MOTION

IN ONE DIMENSION

VARIABLES Time is shown as a little t.“t” is measured in seconds, usually. Sometimes minutes or hours SI unit = seconds (s)

Distance is shown as a little y or a little x.“y” or “x” both mean the same thing. “x” is for horizontal; “y” is for vertical

Usually measured in meters Sometimes kilometers, miles, feet, or inches SI unit = meters (m)

Velocity & speed are shown as little v.SI unit for “v” is meter/second (m/s) Sometimes mph, km/hr, ft/s

WHAT IS MOTION?

Motion is changing position along a certain path.

In the first part of physics, we have 2 goals:To understand how things move (kinematics)To understand why things move (dynamics)

“in one dimension” = horizontal OR vertical

FRAME OF REFERENCE

A reference point is a starting point… a place that everyone calls zero.

When everyone knows where 0 is, then it is much easier to describe the location or position of objects.

Example: In races, the starting line is 0, or your reference point. Everything is measured from the starting line.

POSITION, DISTANCE, AND DISPLACEMENT

All measured with a length unit

Position (x): Where something is located, with respect to a reference point… (not in motion)

When an object is moving…

Distance (d): how far an object travels… magnitude only! (scalar)

Displacement (Δx): an object’s change in position… where an object is with respect to where it startedVector Quantity magnitude AND direction.Only the beginning point and the ending point are important

Final position minus initial positionΔx = xf - xi

DISTANCE VS. DISPLACEMENT

Distance is the total path an object travels. Measures how far an object moves along a path.

Displacement is the line from the beginning to the end.Measures how far between the starting point and ending point.

Start

End

DISTANCE VS. DISPLACEMENT

You drive the path, and your odometer goes up by 8 miles (your distance). Your displacement is the shorter directed distance from start to stop (green arrow). What if you drove in a circle?

start

stop

DISTANCE VS. DISPLACEMENT

You start at your house (0). You go to the store. Then you go to your friend’s house.

What is the distance that you travel?8 houses

What is your displacement from the beginning to the end?2 houses

0 storefriend

8

SPEEDWhen an object is “in motion,” its position changes in time.

Speed describes how fast an object is moving. (scalar)

Speed = distance time

INSTANTANEOUS SPEED – speed at a given moment (or instant) in time

AVERAGE SPEED – Calculated speed for an entire trip

Units for speed include: meters/second (m/s), miles/hour (MPH), feet/minute (ft/min), kilometers/hour (kph), etc..

V = -------X

t

SPEED Example: It takes me roughly 35 minutes to get to school, which is 25 miles away from my house. What’s my average speed?

V = x / t

= 25mi / 35min

= 25mi / 0.58hr

= 43mi/hr

V = -------X

t

SPEED VS. VELOCITY Velocity is an object’s speed in a particular direction. Answers how fast and which way! Velocity is a vector quantity

Average velocity = displacement

time interval

Usually the time interval starts at 0.

The time interval is always positive.

SPEED VS. VELOCITY Speed is a scalar (how fast something is moving regardless of its direction). Ex: v = 20 mph

Speed can be the magnitude of velocity.

Velocity is a combination of speed and direction. Ex: v = 20 mph at 15 south of west

The symbol for speed is v.

The symbol for velocity is type written in bold: v or hand written with an arrow: v

Note:

Average velocity is based on the start and finish and is not focused on what happens in the middle.

Average velocity is the typical value that we calculate when working on simple rate problems.

How do you know when to use speed and when to use velocity?

If the direction doesn’t change, speed and velocity are essentially interchangeable, because speed is the

magnitude of the velocity vector. In most cases, either term will work fine, after we learn the

difference.

SPEED VS. VELOCITY During your 8 mi. trip, which took 15 min., your speedometer displays your instantaneous speed, which varies throughout the trip. Your average speed is 32 mi/hr. Your average velocity is 32 mi/hr in a SE direction. At any point in time, your velocity vector points tangent to your path. The faster you go, the longer your velocity vector.

SPEED VS. VELOCITY You start at your house (0). You go to the store. Then you go to your friend’s house. It takes 10 minutes total.

What is your average speed?8 houses/10 minutes = 0.8 houses/min

What is your average velocity from the beginning to the end?2 houses/10 minutes = 0.2 houses/min

0 storefriend

FUNDAMENTALS OF MOTION

TYPES OF VELOCITY

INSTANTANEOUS VELOCITY – velocity at a given moment (or instant) in time

AVERAGE VELOCITY – calculated velocity for an entire trip

CONSTANT VELOCITY – describes an object that is moving at the same speed and in the same direction (cruise control)

CHANGING VELOCITY – describes a moving object where one of the following occurs:

1) the speed changes 2) the direction changes 3) both change

GRAPHING MOTION To investigate the relationship between distance or displacement and time, you can useMathematical equationsorGraphs of distance vs. time

Dis

tanc

e (m

)

Time (s)

V = -------X

t

HERE IS AN EXAMPLE OF A POSITION VS. TIME GRAPH.

What can it tell us?Where the object was at any time.How long it took the object to travel a certain distance.The velocity of the object. = SLOPE of the line

WHAT ELSE DOES THIS GRAPH TELL US?

It tells us Starting position of each objectEnding position of each objectSpeed and direction of each object

Which object is going fastest

What is B’s speed?1.33m/s

Who is going faster, A or B? A How could we tell that without doing the math?

because the line of B is not as steep as A’s line

A

B

C

What does a flat horizontal line mean?

TimeIncreasing

DistanceStaying the same

SpeedNone. The object is not moving

Dis

tanc

e (m

)

Time (s)

WHAT HAPPENS WHEN SPEED IS NOT CONSTANT?

• This graph shows a constant slope

constant velocity

• Am I traveling 43mi/hr on my entire trip to SKHS?

• No. I did not travel at constant velocity, but, 43mph is an average velocity for 35 minutes. Instead of 35 minutes, I can look at

Every 10 minutes

Every 5 minutes

Every minute

Every second… (taking the limit)

Time (min)

Pos

ition

(m

i)

35

25

Time (min)P

ositi

on (

mi)

35

25

INTERPRETING POSITION VS. TIME

What distance did the object travel in 5 seconds?

What is the object’s displacement after 5 seconds?

What was the object’s average speed?

What was the object’s average velocity?

Can we tell anything else?

Make up a scenario about this object’s adventures in 5 seconds.

EXAMPLE:

24

Vel

oci

ty (

m/s

)

Time (s)

Velocity vs. Time

VELOCITY VS. TIME GRAPH

2

-2 1 2

Slope

A: 4.5m/s

B: 6.75m/s

C: 2.25m/s

D: -2.25m/s

E: -6.75m/s

EXAMPLE:

What does this graph tell us?

What could be happening?

What would the corresponding position vs. time graph look like?

Since the slope of position vs. time tells us velocity, what does the slope of a velocity vs. time graph tell us?

FUNDAMENTALS OF MOTIONACCELERATION:

• The rate at which velocity is changing.

• How fast an object speeds up, slows down, or changes direction.

SI unit = meters/second/second (m/s2). The English unit is feet/second/second (ft/s2)

The word “deceleration” generally indicates a decrease in velocity.

Acceleration is a vector quantity.

The word “acceleration” generally indicates an increase in velocity.

EquationsTo calculate the acceleration, you must divide the

change in velocity by the total time:

t

vv

t

va if

or

time

velocityonaccelerati

In this class we will only use average acceleration… we will keep our accelerations constant!

If our acceleration changes, so does the way we solve our problem… we break it up more!

VELOCITY & ACCELERATION SIGN CHART

V E L O C I T Y ACCELERATION

+ -

+ Moving forward;

Speeding up

Moving backward;

Slowing down

-

Moving forward;

Slowing down

Moving backward;

Speeding up

EXAMPLEYou’re riding a motorcycle at 25 m/s and come to a uniform stop at a red light in 5 seconds. What’s your acceleration?

t

vv

t

v a if

5

25

5

520 a

Given:25 m/s

Looking for: acceleration (a)

5 s

Stop = 0

= vi

= t

= vf

2s

m-5a

GRAPHING W/O ACCELERATION

x

A … Start from rest south of home; head north

B … Pass home; stop (still moving north)

C … Turn around; head south

D … Continue heading south; stop near the starting point

t

A

B C

D

This doesn’t really happen… We are always changing our speed and direction.

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

TANGENT LINES

t

SLOPE DIRECTION

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

CONCAVITY

t

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, R Change of concavity

Peak or Valley Q Turning point

Time Axis Intercept

P, STimes when you are at

“home”

S

CURVE SUMMARY

t

x

Concave Up Concave Down

Increasing v > 0 a > 0 (A)

v > 0 a < 0 (B)

Decreasing

v < 0 a > 0 (D)

v < 0 a < 0 (C)

A

BC

D

ALL 3 GRAPHS

t

x

v

t

a

t

WHY THE PURPLE DOTTED LINES? From a position vs. time to a velocity vs. time: Line up the graphs vertically. Draw vertical dashed lines at special points except intercepts.

Map the slopes of the position graph onto the velocity graph.

A red peak or valley means a blue time intercept.

The same rules apply in making an acceleration graph from a velocity graph. Just graph the slopes!

Note: a positive constant slope in blue means a positive constant green segment. The steeper the blue slope, the farther the green segment is from the time axis.

Everything along the purple dotted lines happens at the same

time!

GRAPH PRACTICETry making all three graphs for the following scenario:

1. Joe 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, Joe 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.

RECAPDisplacement → Δx = xf – xi

Velocity → v =

Acceleration → a =

When talking about horizontal motion, it is usually in a straight line, so displacement = distance.

Δx

ΔtΔvΔt