Chapter 2:
Motion in 1 Dimension
Mars in Retrograde
Describing Motion
Terms to Know:
 Position
 Displacement
 Velocity
 Speed
 Acceleration
Sign Convention:
 “+” to the right
 “-” to the left
Displacement: unit (m), vector
 The displacement of an
object moving along
the x-axis is defined as
the change in position
of the object:
Δx=xf -xi
Where is xi initial position and xf is final position
 Distance is the length of a path followed by a
particle
P
Below is shown a straight track along which a toy
train can move. If the train moves from point A to
point C and then back to point B in 10 seconds, what
is its resulting displacement?
A.+2 meters
B.+3 meters
C.+5 meters
D.+13 meters
E.+15 meters
F.–2 meters
G.–3 meters
H.–5 meters
I.Some other value
meters
Position-Time Graph
 The position-time graph
shows the motion of the
particle (student)
Velocity: unit (m/s), vector
 The velocity of an object during time interval Δt is
the displacement Δx divided by Δt:
𝑣𝑥 =
∆𝑥
∆𝑡
= slope of graph =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
 The speed (scalar) is the magnitude of the velocity and
is always positive
Graphical Interpretation of Velocity:
Draw the car’s velocity vs. time graph
Uniform Motion:
 The velocity is constant for uniform Motion
∆𝑥 𝑥𝑓 − 𝑥𝑖
𝑣𝑥 =
=
∆𝑡
∆𝑡
 position equation for uniform motion
𝑥𝑓 = 𝑥𝑖 + 𝑣𝑥 ∆𝑡
Instantaneous Velocity
 The instantaneous
velocity indicates what
is happening at every
point of time.
 The slope of the
tangent line is the
instantaneous velocity
at that time.
A train car moves along a long straight track.
The graph shows the position as a function of
time for this train. The graph shows that the
train:
A.speeds up all the time.
B.slows down all the time.
C.speeds up part of the time & slows down part of the time.
D.moves at a constant velocity.
From Velocity to position
Displacement is the area under a velocity vs. time graph
𝑥𝑓 = 𝑥𝑖 + 𝑣𝑥 ∆𝑡
A person initially at point P in the illustration stays there
a moment and then moves along the axis to Q and stays
there a moment. She then runs quickly to R, stays there
a moment, and then strolls slowly back to P. Which of
the position vs. time graphs below correctly represents
this motion?
Acceleration: unit (m/s2), vector
The acceleration of an object undergoing a change in
velocity Δv during time interval Δt is: slope of v vs. t graph
∆𝑣𝑥 𝑣𝑥𝑓 − 𝑣𝑥𝑖
𝑎𝑥 =
=
∆𝑡
𝑡𝑓 − 𝑡𝑖
Velocity, Acceleration:
 Velocity and acceleration have the same sign, object is
speeding up
 Velocity and acceleration have the opposite sign, object is
slowing down
The graph shows position as a function of time
for two trains running on parallel tracks. Which is
true?
A.At time tB, both have the same velocity.
B.Both speed up all the time.
C.Both have the same velocity at some time before tB.
D.Somewhere on the graph, both have the same acceleration.
Graphical Comparison:
 Given the x vs. t graph
slope of x vs. t
v vs.t
slope of v vs. t
a vs.t
 Given the a vs. t graph
area of a vs. t
v vs.t
area of v vs. t
x vs.t
 Given the v vs. t graph
area of v vs. t
x vs.t
slope of v vs. t
a vs.t
For the velocity vs. time
graph shown to the right
from 0s to 4s, sketch the
position and acceleration vs.
time graphs. Assume that
the object starts at +5m.
Label the objects final
position and acceleration.
Example Problem 02
1.
For the acceleration vs. time
graph shown to the right from 0s
to 3s, sketch the position and
velocity vs. time graphs. Assume
that the object starts at -5 m with
an initial velocity of 0 m/s. Label
the objects final position and
velocity.
Example Problem Solutions
Graphs shown to the left.
At 3s, v is 6 m/s from area
under a vs. t graph, and x
is 4 m from area under v
vs. t graph.