Motion In One Dimension

PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA

Distance, Position and DisplacementDistance (d)

Distance is the total length of a path traveled by an object.

Distance is always positive, even if an object reverses its direction.

Position (x or y)

Position is the location of an object relative to an origin. Position can be positive or negative.

Displacement (∆x or ∆y)Displacement is the change in position of an object.

Displacement can be positive or negative.

-6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6

CA B

1. What is the distance traveled if an object starts at point C, moves to A, then to B?

2. What are the positions of objects located at points A, B, and C relative to the origin?

3. What is the displacement of an object that starts at point C and moves to point B?

4. What is the displacement of an object that starts at point A, then moves to point C and then moves to point B?

12 units

-5, -2, +4 units

-2 – (+4) = -6 units

-2 – (-5) = +3 units

Distance and Position Graphs

Distance vs. Time Position vs. Timed (m)

t (s)

x (m)

t (s)

Distance graphs show how far an object travels. Speed is determined from the slope of the graph, which can only be positive.

Position graphs show how far, and in which direction, an object travels. Velocity (speed with direction) is determined from the slope of the graph.

constant speed

constantpositive velocity

constantnegative velocity

Notice that these graphs show constant speed. (How do you know?)

posi

tive

neg

ativ

e

Average Speed vs. Average VelocityAverage speed is the distance traveled divided by time elapsed.

average speed =distancetraveled

timeelapsed

€

savg =d

t

Average velocity is displacement divided by time elapsed.

average velocity =displacementtimeinterval

vavg =Δxt

Example: A sprinter runs 100 meters in 10 seconds, and then walks slowly back to the starting blocks in 30 seconds. What is the sprinter’s average speed and average velocity for the entire time?

200

150

100

50

0 40302010

d (

m)

t (s)

200

150

100

50

0 40302010

x (m

)

t (s)

slope = speed

slope = velocity

Instantaneous Speed and VelocityInstantaneous speed is the speed of an object at an exact moment in time. Instantaneous velocity includes direction too.

instananeous speed =distance

time as t approaches zero

Instantaneous speed (or velocity) is found graphically from the slope of a tangent line at any point on a distance (or position) vs. time graph.

instananeous velocity =displacement

time as t approaches zero

slope of tangent =speed

d (

m)

t (s)

x (m

)

t (s)

negative tangent slope =negative velocity

Velocity and Displacement (Honors)

A velocity graph can be used to determine the displacement (change in position) of an object.

The area of the velocity graph equals the object’s displacement.

Velocity vs. Time

v (m

/s)

t (s)

area = displacement

= (.5)(3 s)(30 m/s) + (4 s)(30 m/s) + (.5)(1 s)(30 m/s) = 180 m

30

20

10

08642

For a non-linear velocity graph, the area can be determined by adding up infinitely many pieces each of infinitely small area, resulting in a finite total area!

This process is now known as integration, and the function is called an integral.

The Physics of Acceleration

“Acceleration is how quickly how fast changes”PAUL HEWITT, CITY COLLEGE, S.F.

“how fast”

“how fast changes”

“how quickly”

Acceleration is defined as the rate at which an object’s velocity changes.

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acceleration =change in velocity

timeaavg =

Δvt

means velocity

means change in velocity

mean how much time elapses

Acceleration is considered as a rate of a rate. Why?

Acceleration has units of meters per second per second, or m/s/s, or m/s2.

Metric (SI) units

Types of Acceleration

Constant acceleration is the slope of the line for a velocity vs. time graph.

(Compare to, but DO NOT confuse with constant velocity on a position vs. time graph.)

Average acceleration is the slope of a secant line for a velocity vs. time graph.

Instantaneous acceleration is the slope of a tangent line for a velocity vs. time graph.

(Compare to, but DO NOT confuse with average and instantaneous velocity on a position vs. time graph.)

Constant Acceleration

Velocity vs. Timev (m/s)

t (s)

Velocity vs. Timev (m/s)

t (s)Varying Acceleration

slope

= a

ccel

erat

ion

slope

= a

vera

ge a

ccele

ratio

n

slop

e =

inst

anta

neou

s ac

cele

ratio

n

An Acceleration Analogy.

time (months)

$10.00

$15.00

J F M A M J J A S O N

$20.00

wage vs. time

.

time (sec)

10

15

0 1 2 3 4 5 6 7 8 9 10

20

velocity vs. time

Compare the graph of wage versus time to a velocity versus time graph.

The slope of the wage graph is “wage change rate”. Slope of the velocity graph is acceleration. What is the slope for each graph, including units?

In this case the “wage change rate” is constant. The graph is linear because the rate at which the wage changes is itself unchanging (constant)!

The analogy helps distinguish velocity from acceleration because it is clear that wage and “wage change rate” (acceleration) are different.

slope = acceleration

= 1 m/s/s

slope = “wage change rate”

= $1//hr/month

An Acceleration Analogy

Can a person have a high wage, but a low “wage change rate”?

Earnings, Wage, and “Wage Change Rate” Position, Velocity, and Acceleration

Can an object have a high velocity, but a low acceleration?

Can a person have a positive wage, but a negative “wage change rate”?

Can an object have a positive velocity, but a negative acceleration?

Can a person have zero wage, but still have “wage change rate”?

Can an object have zero velocity, but still have acceleration?

Can a person have a low wage, but a high “wage change rate”?

Can an object have a low velocity, but a high acceleration?

Making good hourly money, but getting very small raises over time.

Moving fast, but only getting a little faster over time.

Making little per hour, but getting very large raises quickly over time.

Moving slowly, but getting a lot faster quickly over time.

Making money, but getting cuts in wage over time.

Moving forward, but slowing down over time.

Making no money (internship?), but eventually working for money.

At rest for an instant, but then immediately beginning to move.

Direction of Velocity and Acceleration

vi a motion

+ 0

– 0

0 +

0 –

+ +

– –

+ –

– +

constant positive vel.

constant negative vel.

speeding up from rest

speeding up from rest

speeding up

slowing down

slowing down

v

t

v

t

v

t

v

t

v

t

v

t

v

t

v

t

speeding up

click for applet

Kinematic Equations of MotionAssuming constant acceleration, several equations can be derived and used to solve motion problems algebraically.

Velocity vs. Time(Constant Acceleration)

v (m/s)

t (s)

v f =vi + at

Δx = vit + 12 at

2

v f2 =vi

2 + 2aΔx

Δx = 12 vi + v f( )t

Slope equals acceleration

a =Δvt

=vf −vi

t⇒

Area equals displacement

A = 12 b1 +b2( )h ⇒

Eliminate time

Eliminate final velocity

vf

vi t

Freefall AccelerationAristole wrongly assumed that an object falls at a rate proportional to its weight.

Galileo proved that all objects freefall (in a vacuum, no air resistance) at the same rate.

An inclined plane reduced the effect of gravity, showing that the displacement of an object is proportional to the square of time.

Kinematic equations of freefall acceleration:

vyf =vyi + gt

Δy = vyit + 12 gt

2

vyf2 =vyi

2 + 2gΔy

Δy = 12 vyi + vyf( )t

Since the acceleration is constant, velocity is proportional to time.

Δy : t 2

Location g

Equator 9.780

Honolulu 9.789

Denver 9.796

San Francisco 9.800

Munich 9.807

Leningrad 9.819

North Pole 9.832

click for video

Latitude, altitude, geology affect g.

A Velocity AnalogyCompare constant velocity (uniform motion) to making money doing a job.

Say you baby sit for $10/hour. It’s easy to graph earnings as a function of time.

Compare the graph of earnings versus time to position versus time.

The slope of the earnings graph is wage. Slope of the position graph is velocity.

.

earnings vs. time

time (hours)5 10

$50

$100

0

.

position vs. time

time (seconds)5 10

50

100

0

slope = velocity

= 10 m/s

slope = wage

= $10//hr

.

time (months)J F M A M J J A S O N

earnings vs. time

.

position vs. time

time (sec)0 1 2 3 4 5 6 7 8 9 10

A Velocity AnalogyCompare constant acceleration to getting raises while doing a job.

Maybe you baby sit for $10/hour, but now you get regular wages increases.

Compare the graph of earnings versus time to position versus time.

Slope of a secant line is the average wage (compare with average velocity.)

Slope of a tangent line is the instantaneous wage (compare with instant. velocity.)