1/23/14 Agenda
•  We need a note-taker! If you’re interested,
see me after class.
•  Today: HW Quiz #1, 1D Motion
•  Lecture for this week: Chapter 2 (finish
reading Chapter 2 by Thursday)
•  Homework #2: continue to check the class
website/schedule page
Quantities in Motion
•  Any motion involves three concepts
–  Displacement (x, m)
–  Velocity (v, m/s)
–  Acceleration (a, m/s2)
•  These concepts can be used to study
objects in motion
Speed
•  The average speed of an object is defined as
the total distance traveled divided by the total
time elapsed
Average speed =
d
v=
t
–  Speed is a scalar quantity
total distance
total time
Chapter 1 Problem 28:
A city has streets laid out in a square grid,
with each block 135 m long. If you drive
north for three blocks, then west for three
blocks, how far are you from your starting
point? Show all your work.
Displacement Isn’t Distance
•  The displacement of an object is not
the same as the distance it travels
– Example: Throw a ball straight up and
then catch it at the same point you
released it
•  The distance is twice the height
•  The displacement is zero
Velocity
•  It takes time for an object to undergo a
displacement
•  The average velocity is rate at which the
displacement occurs
v average =
Δx x f − x i
=
Δt
tf − ti
•  generally use a time interval, so let ti = 0
1 1/23/14 Velocity continued
Speed vs. Velocity
•  Direction will be the same as the direction
of the displacement (time interval is
always positive)
–  + or - is sufficient
•  Units of velocity are m/s
•  Cars on both paths have the same average velocity
since they had the same displacement in the same
time interval
•  The car on the blue path will have a greater
average speed since the distance it traveled is
larger
Distance & Displacement LectureTutorial
•  Work with a partner or two
•  Read directions and answer all questions
carefully. Take time to understand it now!
•  Come to a consensus answer you all agree
on before moving on to the next question.
•  If you get stuck, ask another group for help.
•  If you get really stuck, raise your hand and I
will come around.
Average Velocity, Constant
•  The straight (not
curved) line
indicates constant
velocity
•  The slope of the line
is the value of the
average velocity
Graphical Interpretation of Velocity
•  Velocity can be determined from a
position-time graph
•  Average velocity equals the slope of the
line joining the initial and final positions
•  An object moving with a constant velocity
will have a graph that is a straight line
Average Velocity, Non Constant
•  This motion is
non-constant
velocity
•  The average
velocity is the
slope of the blue
line joining two
points
2 1/23/14 Instantaneous Velocity
•  The limit of the average velocity as the time
interval becomes infinitesimally short, or as the
time interval approaches zero
lim Δx
v ≡Δt→0
Δt
•  The instantaneous velocity indicates what is
happening at every point of time
Uniform Velocity
•  Uniform velocity  constant velocity
•  The instantaneous velocities are always
the same
–  All the instantaneous velocities will also equal
the average velocity
Instantaneous Velocity on a Graph
•  The slope of the line tangent to the
position-vs.-time graph is defined to be the
instantaneous velocity at that time
–  The instantaneous speed is defined as the
magnitude of the instantaneous velocity
Acceleration
•  Changing velocity (non-uniform) means an
acceleration is present
•  Acceleration is the rate of change of the
velocity
a=
•  Units are m/s²
Δv vf − vi
=
Δt
tf − ti
Agenda
Acceleration
•  Today: Finish Chapter 2 (Motion graphs,
kinematics equations, and free fall)
•  Reading for next week: Chapter 3
•  Still no change in enrollment
•  HW #1/quiz can be picked up at the end of
class
•  Vector quantity: rate of change of velocity
•  When the sign of the velocity and the
acceleration are the same (either positive
or negative), then the speed is increasing
•  When the sign of the velocity and the
acceleration are in the opposite directions,
the speed is decreasing
3 1/23/14 Relationship Between
Acceleration and Velocity
•  Uniform velocity (shown by red arrows
maintaining the same size)
•  Acceleration equals zero
Relationship Between Velocity
and Acceleration
•  Acceleration and velocity are in opposite directions
•  Acceleration is uniform (blue arrows maintain the
same length)
•  Velocity is decreasing (red arrows are getting
shorter)
•  Velocity is positive and acceleration is negative
The three graphs in the figure represent
the position vs. time for objects moving
along the x-axis. Which, if any, of these
graphs is not physically possible?
Relationship Between Velocity
and Acceleration
•  Velocity and acceleration are in the same direction
•  Acceleration is uniform (blue arrows maintain the
same length)
•  Velocity is increasing (red arrows are getting
longer)
•  Positive velocity and positive acceleration
Acceleration Lecture-Tutorial
•  Work with a partner or two
•  Read directions and answer all questions
carefully. Take time to understand it now!
•  Come to a consensus answer you all agree
on before moving on to the next question.
•  If you get stuck, ask another group for help.
•  If you get really stuck, raise your hand and I
will come around.
Graphical Interpretation of
Acceleration
•  Average acceleration is the slope of the
line connecting the initial and final
velocities on a velocity-time graph
•  Instantaneous acceleration is the slope of
the tangent to the curve of the velocitytime graph
4 1/23/14 Average Acceleration
Kinematic Equations
•  Used in situations with uniform
acceleration
v = vo + at
1
v +v t
2 o
1
Δx = vot + at 2
2
v 2 = vo2 + 2aΔx
Δx = vt =
(
)
Notes on the equations
v = vo + at
•  Shows velocity as a function of
acceleration and time
•  Use when you don’t know and aren’t
asked to find the displacement
Parts (a), (b), and (c) of the figure represent three
graphs of the velocities of different objects moving in
straight-line paths as functions of time. The possible
accelerations of each object as functions of time are
shown in parts (d), (e), and (f). Match each velocity
vs. time graph with the acceleration vs. time graph
that best describes the motion.
Notes on the equations
Δx = v o t +
1 2
at
2
•  Gives displacement as a function of time,
velocity and acceleration
•  Use when you don’t know and aren’t
asked to find the final velocity
Notes on the equations
& v + vf #
Δx = v average t = $ o
!t
% 2 "
•  Gives displacement as a function of
velocity and time
•  Use when you don’t know and aren’t
asked for the acceleration
•  Careful! vavg is not the same as vi or vf
5 1/23/14 Notes on the equations
Free Fall
v 2 = vo2 + 2aΔx
•  All objects moving under the influence of
gravity only are said to be in free fall
•  Gives velocity as a function of acceleration
and displacement
•  Use when you don’t know and aren’t
asked for the time
–  Free fall does not depend on the object’s
original motion
Acceleration due to Gravity
•  Symbolized by g
•  g = 9.80 m/s²
–  When estimating, use g ≈ 10 m/s2
•  g is always directed downward
–  toward the center of the earth
•  Ignoring air resistance and assuming g
doesn’t vary with altitude over short
vertical distances, free fall is constantly
accelerated motion
•  All objects falling near the earth’s surface
fall with a constant acceleration
•  The acceleration is called the acceleration
due to gravity, and indicated by g
Free Fall – an object dropped
•  Initial velocity is zero
•  Let up be positive
•  Use the kinematic
equations
–  Generally use y
instead of x since
vertical
vo= 0
a = -g
•  Acceleration is -g =
-9.80 m/s2
6