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Motion In One Dimension

PLATO AND ARISTOTLE

GALILEO GALILEI

LEANING TOWER OF PISA

Distance, Position and Displacement

Distance (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.

1. What is the distance traveled if an object

starts at point C, moves to A, then to B?

12 units

Displacement (?x or ?y)

Displacement is the change in position of an

object.

2. What are the positions of objects located at

points A, B, and C relative to the origin?

-5, -2, 4 units

3. What is the displacement of an object that

starts at point C and moves to point B?

-2 (4) -6 units

4. What is the displacement of an object that

starts at point A, then moves to point C and then

moves to point B?

Displacement can be positive or negative.

-2 (-5) 3 units

Distance and Position Graphs

Distance vs. Time

Position vs. Time

d (m)

x (m)

positive

constant positive velocity

t (s)

constant speed

negative

constant negative velocity

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.

Notice that these graphs show constant speed.

(How do you know?)

Average Speed vs. Average Velocity

Average speed is the distance traveled divided by

time elapsed.

Average velocity is displacement divided by time

elapsed.

Example A sprinter runs 100 meters in 10

seconds, and then walks slowly back to the

starting blocks in 30 seconds. What is the

sprinters average speed and average velocity for

the entire time?

x (m)

d (m)

slope speed

slope velocity

t (s)

t (s)

Instantaneous Speed and Velocity

Instantaneous speed is the speed of an object at

an exact moment in time. Instantaneous velocity

includes direction too.

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.

negative tangent slope negative velocity

x (m)

d (m)

slope of tangent speed

t (s)

t (s)

Velocity and Displacement (Honors)

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

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.

A velocity graph can be used to determine the

displacement (change in position) of an

object. The area of the velocity graph equals the

objects displacement.

The Physics of Acceleration

Acceleration is how quickly how fast changes

PAUL HEWITT, CITY COLLEGE, S.F.

how fast

means velocity

how fast changes

means change in velocity

how quickly

mean how much time elapses

Acceleration is defined as the rate at which an

objects velocity changes.

Metric (SI) units

Acceleration has units of meters per second per

second, or m/s/s, or m/s2.

Acceleration is considered as a rate of a rate.

Why?

Types of Acceleration

Velocity vs. Time

Velocity vs. Time

v (m/s)

v (m/s)

slope average acceleration

slope acceleration

slope instantaneous acceleration

t (s)

t (s)

Constant Acceleration

Varying Acceleration

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 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.)

An Acceleration Analogy

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

Earnings, Wage, and Wage Change Rate

Position, Velocity, and Acceleration

Can a person have a high wage, but a low wage

change rate?

Can an object have a high velocity, but a low

acceleration?

Making good hourly money, but getting very small

raises over time.

Moving fast, but only getting a little faster

over time.

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 little per hour, but getting very large

raises quickly over time.

Moving slowly, but getting a lot faster quickly

over time.

Can a person have a positive wage, but a negative

wage change rate?

Can an object have a positive velocity, but a

negative acceleration?

Making money, but getting cuts in wage over time.

Moving forward, but slowing down over time.

Can a person have zero wage, but still have wage

change rate?

Can an object have zero velocity, but still have

acceleration?

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

speeding up

slowing down

slowing down

click for applet

Kinematic Equations of Motion

Assuming constant acceleration, several equations

can be derived and used to solve motion problems

algebraically.

Slope equals acceleration

Velocity vs. Time (Constant Acceleration)

v (m/s)

Area equals displacement

vf

vi

t

Eliminate final velocity

t (s)

Eliminate time

Freefall Acceleration

Aristole 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.

click for video

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

Kinematic equations of freefall acceleration

Since the acceleration is constant, velocity is

proportional to time.

Latitude, altitude, geology affect g.

A Velocity Analogy

Compare constant velocity (uniform motion) to

making money doing a job. Say you baby sit for

10/hour. Its 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.

slope velocity 10 m/s

slope wage 10//hr

A Velocity Analogy

Compare 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.)