Lecture 3: 1-D Kinematics
This Week’s Announcements:
• Class Webpage:
•  http://kestrel.nmt.edu/~dmeier/phys121/phys121.html
•  visit regularly
•  Our TA is Lorraine Bowman
Week 2 Reading:
Chapter 2 - Giancoli
Due: 9/1
Week 2 Assignments:
- HW #1 (chp 1: Q5, P2, P8, P39 & chp 2: Q2, Q18, P7, P9a-d, P35, P42)
- for this one assignment you must report your answers with proper units and sig.
figs. for full credit !!
- MasteringPhysics: -Introduction to Mastering Physics (if not done
yet)
This Week’s Goal:
- Define and introduce the basic kinematic variables, such as
displacement, velocity and acceleration, and begin applying them to
one dimensional motion.
Wee! I’m learning kinematics
space (x)
Space-time diagrams
time
-  Graph shows the evolution of the position (1-D) in time (how things
move in time) --- not (x vs. y) but x vs. time
Distance and Displacement
+
- Position: The spatial coordinate of
an object as a function of time [x].
-generally have the freedom to select the
zero of our coordinate system as we like
- Distance: The length covered
along a path.
Xto = +1 mil
-always a positive quantity
xfrom = -1 mil
- Displacement: The net change in
position (and direction) for a path
[Δx = x - xo].
-Displacement of a round trip is
zero !
X =0
-
- Keep to 1-D for the moment:
1 mil
space (x)
Introduction to space-time diagrams
20 m 30 m
-  Space-time diagram for trip to Little Caesar’s
-(walking at a constant rate, and waiting in line)
50 m time
Clicker Question:
3) Which of the following space-time diagrams represent an object which
ultimately comes to a stop and remains stopped ?
x
a)
x
t
b)
x
t
c)
x
t
d)
x
t
e)
t
Clicker Question:
3) Which of the following space-time diagrams represent an object which
ultimately comes to a stop and remains stopped ?
x
a)
x
t
b)
x
t
c)
x
t
d)
x
t
e)
t
Velocity and Speed
- Average speed, s:
(distance traveled)/(total time)
x
Slope = rise/run
= Δx/Δt
x2
Δx = x2 - x1
-always a positive quantity
x1
- Average velocity:
(displacement)/(total time)
-includes direction information
t1
t2
Δt = t2 - t1
t
x
- instantaneous velocity, v, (or
simply “velocity”): (The change in
displacement over a very tiny time
interval) /(that tiny time interval)
dx
dt
-includes direction information
x(t+h) - x(t) = Δx
- instantaneous speed, v, (or
simply “speed”): The magnitude
of the instantaneous velocity.
-always a positive quantity
h = Δt
t
Slope = rise/run = Δx/Δt
V = lim
h
0
x(t+h) - x(t)
h

V = dx/dt
-This is precisely the definition of the time derivative of x.
Calculus Digression
x(t) = t
dx(t)/dt =1
Slope = 2/2 = 1
*
Slope = 2/2 = 1
*
2
2
*
Slope = rise/run
Slope over infinitesimal distance=dx(t)/dt
2
V = lim
h
0
x(t+h) - x(t)
h
2
x(t) = t2
2
2
=
dx
dt
dx(t)/dt =2t
Slope = 2/2 = 1
Slope = 4/1 = 4
- When the curve is
horizontal, Δx is zero, so
the slope Δx/Δt = 0
-setting the derivative
to zero finds the local
minima or maxima
∫ = integral = “anti-derivative”
* 4
1
-2 *
Slope = -2/1 = -2
1
2
*
*
1
Slope = 0/1 = 0
1
Slope = 2/1 = 2
Clicker Question:
4) The following is a space time diagram for an object moving in 1-D:
x
t
Which of the following represents the velocity vs. time diagram for the object?
v
v
t
a)
v
t
b)
v
t
c)
v
t
d)
t
e)
Clicker Question:
4) The following is a space time diagram for an object moving in 1-D:
x
t
Which of the following represents the velocity vs. time diagram for the object?
v
v
t
a)
v
t
b)
v
t
c)
v
t
d)
t
e)
Acceleration
- Acceleration is the change
in velocity with time, in a
completely analogous
fashion to velocity being a
change in displacement with
time
v
dv
dt
- average acceleration:
(change in velocity)/(total time)
v(t+h) - v(t)
-includes direction information
- instantaneous acceleration, a, (or
simply “acceleration”): (The
change in velocity over a very tiny
time interval) /(that tiny time
interval)
-includes direction information
h
v(t+h) - v(t)
a = lim
h 0
h
t
=
d 2x
dv
dt = dt2 = a
-This is precisely the definition of the
time derivative of v.
Velocity
x
(slope on space-time diagram)
dx
dt
Acceleration
(slope on velocity-time diagram)
v
t
dv
dt
t
Clicker Question:
5) The following is a velocity vs. time diagram for an object moving in 1-D:
v
t
Which of the following represents the acceleration vs. time diagram for the object?
a
a
t
a)
a
t
b)
a
t
c)
a
t
d)
t
e)
Clicker Question:
5) The following is a velocity vs. time diagram for an object moving in 1-D:
v
t
Which of the following represents the acceleration vs. time diagram for the object?
a
a
t
a)
a
t
b)
a
t
c)
a
t
d)
t
e)
x
t
Example:
Non constant acceleration
v
t
a
t
Clicker Question:
x
6) Which of the following represents the
time rate of change of the
acceleration (d3x/dt3) [aka the ‘jerk’]?
v
+
(d3x/dt3)
t
-
e)
(d3x/dt3)
+
-
t
t
+
a
(d3x/dt3)
d)
t
t
-
-
c)
(d3x/dt3)
+
b)
(d3x/dt3)
-
a)
+
t
t
t
Clicker Question:
x
6) Which of the following represents the
time rate of change of the
acceleration (d3x/dt3) [aka the ‘jerk’]?
v
+
(d3x/dt3)
t
-
e)
(d3x/dt3)
+
-
t
t
+
a
(d3x/dt3)
d)
t
t
-
-
c)
(d3x/dt3)
+
b)
(d3x/dt3)
-
a)
+
t
t
t
Motion diagrams
1-D:
velocity
- speeding up: acceleration present
x
- acceleration sign (+) if velocity
increases in direction of increasing
coordinate
- slowing down: acceleration present
velocity
x
-  A graph the represents the
location of an object at a series of
equal spaced steps in time
- acceleration sign (-) if velocity
decreases in direction of increasing
coordinate
2-D:
y
velocity
x
General Definitions:
x(t) = the position of an object as a function of time
v(t) =
a(t) =
dv
dt
=
dx
dt
=
(
)
dt dt
d
dx
-Orv = ∫ a dt
x = ∫ v dt = ∫ [∫a dt] dt
d 2x
dt2