Motion In One Dimension

PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA

Distance vs. Time for Toy Car (0-5 sec.)

Distance (cm)

Time (s)0

200

400

600

800

1000

1.0 2.0 3.0 4.0 5.0

Constant speed is the slope of the (best fit) line for a distance vs. time graph.

speed =distancetime

best-fit line (from TI calculator)d =207.7 × t−12.6

s =208 cms

(3 sig figs)

s =208 cms

×10−2 m1 cm

=2.08 ms

Remember, the standard metric unit for length is the meter!

Graphing Constant Speed

Distance vs. Time for Toy Car (0-0.5 sec.)

Distance (mm)

Time (s)0

100

200

300

400

500

0.1 0.2 0.3 0.4 0.5

Average speed is the slope of a secant line for a distance vs. time graph.average speed =

distancetime

600

(0.13, 0)

(0.5, 350)

Instantaneous speed is the slope of a tangent line for a distance vs. time graph.inst. speed =

distancetime

(as t→ 0)

click for applet

savg =1190 m m

s×10−3 m

1 m m= 1.19

m

s

savg =500−25 m m0.5−0.1 s

s=350−0 m m0.5−0.13 s

s=946 m ms

×10−3 m1 m m

= 0.946 ms

Graphing Average and Instantaneous Speed

approximate slopeat t =0.25 s

best-fit quadratic (from TI calculator)

d =2054t2 −26.8t+1.8

tanget line slope at t =0.25 (from calc.)d =1000t−126.6

Distance, Position and DisplacementDistance (d)

The length of a path traveled by an object. It is never negative, even if an object reverses its direction.

Position (x or y)The location of an object relative to an origin. It can be either positive or negative

Displacement (∆x or ∆y)The change in position of an object. Also can be positive or negative.

Δx = x f − xiΔy = y f − yi

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

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

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

d = d1 + d2 = 9 + 3 = 12 m

xf – xi = -2 – (+4) = -6 m

xf – xi = -2 – (-5) = +3 m

One dimensional motion

4. What is the distance traveled and the displacement of the person that starts at point A, then moves to point B, and ends at point C?

Two dimensional motion

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

CA B x(m)

d = 3 + 4 = 7 m; x = √32 + 42 = 5 m

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 cannot be negative. Position graphs show initial position, displacement, velocity (magnitude and direction). That’s why position graphs are better!

CAR A:constant speed

CAR B:constant

positive velocity

CAR C:constant

negative velocity

Remember, all of these graphs show constant speed. (How do you know?)

posi

tive

nega

tive

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

average speed =distance traveledtime elapsed

€

savg =dt

Average velocity is displacement divided by time elapsed.

average velocity =displacementtime interval vavg =

Δxt

Example: A sprinter runs 100 m in 10 s, jogs 50 m further in 10 s, and then walks back to the finish line in 20 seconds. What is the sprinter’s average speed and average velocity for the entire time?

savg =dt=200 m40 s

=5 ms

vavg =Δxt

=100 m40 s

=2.5 ms

200

150

100

50

0 40302010

d (m

)

t (s)

200

150

100

50

0 40302010

x (m

)t (s)

slope = ave. speedslope = ave. velocity

Instantaneous Speed and VelocityInstantaneous speed is the how fast an object moves at an exact moment in time. Instantaneous velocity has speed and direction.

instananeous speed =distancetime

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 =instantaneous speed

d (m

)

t (s)

x (m

)

t (s)

slope of tangent =instantaneous velocitysign of slope =sign of velocity

s=limt→ 0

dt

v=limt→ 0

Δxt

=dxdt

Honors:

The Physics of Acceleration“Acceleration is how quickly how fast changes”

“how fast”“how fast changes”“how quickly”

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

€

acceleration =change in velocity

timeaavg =

Δvt

means velocitymeans 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. m

ss

or ms2

Metric (SI) units

Types of Acceleration

Constant acceleration is the slope of a velocity vs. time graph.(Sound familiar?! 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.(Again, 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

ccele

ratio

n

slope

= av

erage

acce

lerati

on

slope

= in

stan

tane

ous

acce

lera

tion

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

3020100 8642

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.

An Acceleration AnalogyCompare 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 = “wage change rate” = $1//hr/month

slope = acceleration = 1 m/s/s

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 AccelerationCan 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

Velocity vs. Time

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 assumed 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.

v : t

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.