Define Position, Displacement

1
Lecture 2

• Chapter 2.1-2.4
• Define Position, Displacement Distance
• Distinguish Time and Time Interval
• Define Velocity (Average and Instantaneous),
  Speed
• Define Acceleration
• Understand algebraically, through vectors, and
  graphically the relationships between position,
  velocity and acceleration
• Comment on notation

2
Informal Reading Quiz

• Displacement, position, velocity acceleration
  are the main quantities that we will discuss
  today.
• Which of these 4 quantities have the same units
• Velocity position
• Velocity acceleration
• Acceleration displacement
• Position displacement
• Position acceleration

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Range of Lengths

• Distance Length (m)
• Radius of Visible Universe 1 x 1026
• To Andromeda Galaxy 2 x 1022
• To nearest star 4 x 1016
• Earth to Sun 1.5 x 1011
• Radius of Earth 6.4 x 106
• Willis Tower 4.5 x 102
• Football Field 1 x 102
• Tall person 2 x 100
• Thickness of paper 1 x 10-4
• Wavelength of blue light 4 x 10-7
• Diameter of hydrogen atom 1 x 10-10
• Diameter of proton 1 x 10-15

4
Range of Times

• Interval Time (s)
• Age of Universe 5 x 1017
• Age of Grand Canyon 3 x 1014
• Avg age of college student 6.3 x 108
• One year 3.2 x 107
• One hour 3.6 x 103
• Light travel from Earth to Moon 1.3 x 100
• One cycle of guitar A string 2 x 10-3
• One cycle of FM radio wave 6 x 10-8
• One cycle of visible light 1 x 10-15
• Time for light to cross a proton 1 x 10-24

Worlds most accurate timepiece Cesium fountain
Atomic Clock Lose or gain one second in some 138
million years
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One-Dimension Motion (Kinematics) Position,
Displacement, Distance

• Position Reflects where you are.
• KEY POINT 1 Magnitude, Direction, Units
• KEY POINT 2 Requires a reference point
  (Origin)
• Origins are arbitrary

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One-Dimension Motion (Kinematics) Position,
Displacement, Distance

• Position Reflects where you are.
• KEY POINT 1 Magnitude, Direction, Units
• KEY POINT 2 Requires a reference point
  (Origin)
• Origins are arbitrary

Boston
Example Where is Boston ? Choose origin at
New York Boston is 212 miles northeast of New
York OR Boston is 150 miles
east and 150 miles north of New York
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One-Dimension Motion (Kinematics) Position,
Displacement, Distance

• Getting from New York to Boston requires a PATH
• Path defines what places we pass though
• Displacement Change in position
• Requires a time interval
• Any point on the path must be
• associated with a specific time
• ( t1, t2, t3, .)
• Path 1 and Path 2 have the
• same change in position so they
• the same displacement.
• However the distance travelled is
• different.

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Motion in One-Dimension (Kinematics) Position

• Position along a line references xi and ti
• At time 0 seconds Pat is 10 meters to the
  right of the lamp
• Origin ? lamp
• Positive direction ? to the right of the lamp
• Position vector ( xi , ti) or (10 m, 0.0 s)
• Particle representation

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Displacement

• One second later Pat is 15 meters to the right of
  the lamp
• At t 1.0 s the position vector is ( xf , tf )
  or (15 m, 1.0 s)
• Displacement is just change in position
• ?x xf xi
• There is also a change in time
• ?t tf ti

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Displacement

• Putting it all together
• ?x xf - xi 5 meters to the right !
• ?t tf - ti 1 second
• Relating ?x to ?t yields average velocity

O
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Average VelocityChanges in position vs Changes
in time

• Average velocity displacement per time
  increment , includes BOTH magnitude and direction

• Pats average velocity was 5 m / s to the right

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Average Speed

• Average speed, vavg, reflects a magnitude
• How fast without the direction.
• References the total distance travelled

• Pats average speed was 5 m / s
• NOTE Serways notation varies from other texts
• (There really is no standard)

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Pat on tour (graphical representation)

• Pat is walking from and to the lamp (at the
  origin).
• (x1 , t1) (10 m, 0.0 sec)
• (x2 , t2) (15 m, 1.0 sec)
• (x3 , t4) (30 m, 2.0 sec)
• (x4 , t4) (10 m, 3.0 sec)
• (x5 , t5) ( 0 m, 4.0 sec)
• Compare displacement distance avg. vel.
  avg. speed
• t 1 s ?x1,2 x2 x1 5 m d 5 m
  vx,avg 5 m/s vx,avg 5 m/s
• t 2 s ?x1,3 x3 x1 20 m d 20 m
  vx,avg 10 m/s vx,avg 10 m/s
• t 3 s ?x1,4 x4 x1 0 m d 40 m
  vx,avg 0 m/s vx,avg 13 m/s
• Here d ?x1,2 ?x2,3 ?x3,4 5 m 15
  m 20 m 40 m
• Speed and velocity measure different things!

t (seconds)
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Calculating path distance in general
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Exercise 2 Average Velocity
x (meters)
6
4
2
0
t (seconds)
1
2
4
3
-2
What is the magnitude of the average velocity
over the first 4 seconds ?
(A) -1 m/s
(D) not enough information to decide.
(C) 1 m/s
(B) 4 m/s
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Average Velocity Exercise 3 What is the average
velocity in the last second (t 3 to 4) ?
x (meters)
6
4
2
-2
t (seconds)
1
2
4
3

1. 2 m/s
2. 4 m/s
3. 1 m/s
4. 0 m/s

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Average Speed Exercise 4What is the average
speed over the first 4 seconds ?0 m to -2 m to 0
m to 4 m ? 8 meters total

1. 2 m/s
2. 4 m/s
3. 1 m/s
4. 0 m/s

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Instantaneous velocity

• Limiting case as the change in time ? 0

x

• Yellow lines are
• average velocities

instantaneous velocity at t 0 s
t
0

• As Dt ? 0 velocity is the tangent to the curve (
  path)
• Dashed green line is vx

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Instantaneous speed

• Just the magnitude of the instantaneous velocity

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Exercise 5Instantaneous Velocity
x (meters)
6
4
2
-2
t (seconds)
1
2
4
3
What is the instantaneous velocity at the fourth
second?
(B) 0 m/s
(A) 4 m/s
(D) not enough information to decide.
(C) 1 m/s
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Special case Instantaneous velocity is constant

• Slope is constant over a time Dt.

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Special case Instantaneous velocity is constant

• Slope is constant over a time Dt.

x
(xf , tf)
Dx
(xi , ti)
t
Dt
0

• Given Dt, xi and vx we can deduce xf
• and this reflects the area under the velocity
  curve

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Now multiple vx Pats velocity plot

• (x1 , t1) (10 m, 0.0 s)
• (x2 , t2) (15 m, 1.0 s)
• (x3 , t3) (30 m, 2.0 s)
• (x4 , t4) (10 m, 3.0 s)
• (x5 , t5) ( 0 m, 4.0 s)

t (seconds)
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Home exercise 6 (and some things are easier than
they appear)

• A marathon runner runs at a steady 15 km/hr. When
  the runner is 7.5 km from the finish, a bird
  begins flying from the runner to the finish at 30
  km/hr. When the bird reaches the finish line, it
  turns around and flies back to the runner, and
  then turns around again, repeating the
  back-and-forth trips until the runner reaches the
  finish line.
• How many kilometers does the bird travel?

A. 10 km B. 15 km C. 20 km D. 30 km
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Objects with slowly varying velocities
x
t

• Change of the change.
• changes in velocity with time
• give average acceleration

vx
Dt
Dvx
t
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And finally instantaneous acceleration
t
ax
t
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Example problem

• A car moves to the right first for 2.0 sec at 1.0
  m/s and then 4.0 seconds at 2.0 m/s.
• What was the average velocity?

• Two legs with constant velocity but .

Slope of x(t) curve
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Example problem

• A particle moves to the right first for 2.0
  seconds at 1.0 m/s and then 4.0 seconds at 2.0
  m/s.
• What was the average velocity?

• Two legs with constant velocity but .
• We must find the total displacement (x2 x0)
• And x1 x0 v0 (t1-t0) x2 x1 v1
  (t2-t1)
• Displacement is (x2 - x1) (x1 x0) v1
  (t2-t1) v0 (t1-t0)
• x2 x0 1 m/s (2 s) 2 m/s (4 s) 10 m in
  6.0 s or 1.7 m/s

Slope of x(t) curve
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Position, velocity acceleration

• All are vectors!
• Cannot be used interchangeably (different units!)
• (e.g., position vectors cannot be added directly
  to velocity vectors)
• But the directions can be determined
• Change in the position vector vs. time gives
  the direction of the velocity vector
• Change in the velocity vector vs. time gives
  the direction of the acceleration vector
• Given x(t) ? vx(t) ? ax (t)
• Given ax (t) ? vx (t) ? x(t)

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Assignment

• Reading for Tuesdays class
• All of Chapter 2