Chapter 2: Motion in One Dimension
DISPLACEMENT AND VELOCITY
To describe motion, we must first specify ____________________. But to specify position, we need to
relate it to __________________________________________.
For motion in one dimension, it is convenient to use the _______________.
From this concept, we can create a number line.
Displacement of an object –
NOTE: displacement is not always equal to the total distance traveled.

where, x =
xf=
xi=
Example: If a car moves from an initial position of 10 m to a final position of 80 m:
Example: If a car moves from an initial position of 80 m to a final position of 20m:
Note: the negative sign indicates the direction of displacement.
Example: Every morning you drive __ miles from home to NPHS and then come back home at night using
the same route. What is the best statement describing your daily trip? (Assume you do not move far
while at NPHS) displacement = ______ and distance traveled = _____
VELOCITY –
Example: A train leaves Denver at 8:05 a.m. and travels directly to Colorado Springs, arriving at 9:02 a.m.
The distance between Denver and Colorado Springs is _______ miles. Find the average velocity in
miles/hr of the train.
ACCELERATION –
Units – acceleration has units of velocity/time, or…
Time Graphs
Both displacement and velocity can be graphed as a function of time
Example 1
Position
(m)
Example 2
Position
(m)
Time (sec)
Time (sec)
Example 3
Position
(m)
Example 4
Position
(m)
Time (sec)
Time (sec)
SPEED vs. VELOCITY
Both describe how fast the _______________ is changing with respect to _________.
Speed is a _____________ quantity. It indicates ________________ (magnitude), but not direction.
Velocity is a ____________ quantity. It indicates both ____________ (magnitude) and ___________.
SLOPE REVIEW
slope =
+ slope:
-slope:
0 slope:
slope undefined:
VELOCITY
Velocity is represented by the _____________ of the curve on a displacement vs. time graph.
In this class, a positive (+) slope indicates a forward direction and a negative (-) slope indicates a
backwards direction (return).
Slope of a position vs. time graph =
INSTANTANEOUS VELOCITY
Position
(m)
Position
(m)
Time (sec)
The slope of a curve at a given point is equal to the slope
slope of a tangent line at that point. The slope of this
line represents instantaneous velocity at the indicated
point.
Example: The x(t) graph describes a 1-D motion of a train. What must be true about this motion?
Time (sec)
Example: The x(t) graph displays motions of two trains A and B on parallel tracks. Which statement is
true?
ACCELERATION
Motion can be described with a velocity vs. time graph.
Velocity
(m/s)
Velocity
(m/s)
Time (sec)
Time (sec)
For a velocity vs time graph:
Slope =
Velocity
(m/s)
Time (sec)
Example: A train travels at 5 miles per hour for 1 hour.
What is its displacement after 1 hour?
Displacement can also be determined by finding
the area under the curve (AUC) of a velocity
vs. time graph.
Velocity vs. Time
Velocity
(m/s)
Time (sec)
Graphical analysis summary:
Displacement vs. Time graph:
Velocity vs. Time graph:
Drawing V vs. T graphs: A blue car moving at a constant speed of 10 m/s passes a red car that is at rest.
This occurs at a stoplight the moment that the light turns green.. The clock is reset to 0 seconds and the
velocity-time data for both cars is collected and plotted. The red car accelerates from rest at 4 m/s/s
for three seconds and then maintains a constant speed. The blue car maintains a constant speed of 10 m/s
for the entire 12 seconds.
Quick Matching Quiz: Match the velocity graphs with their acceleration graphs
Motion Maps:
* Imagine a toy car traveling along a piece of
paper and dropping a dot of ink at a given time
interval (say 1 drop every second). It could
produce a trail that looks like this:
* How long does it take the car to travel the length of the paper?
* Describe the car’s motion:
Example: Draw the motion map for the following complex motion.: Object accelerates for 3 seconds. Then
travels at a constant velocity for 2 seconds. Then decelerates for 3 seconds. Then returns to the start in
4 seconds at a constant velocity.
VELOCITY WITH CONSTANT ACCELERATION
slope =
Velocity
(m/s)
Time (sec)
Find the AUC…
5 Parameters of Motion:
1. a =
2. x=
3. vf=
4. vi=
5. t=
To solve a constant acceleration problem, you must know, or be able to find, three of the five parameters.
Then use the following equations to solve for the other two:
variable missing:
vf = vi + at
x = ½(vi + vf)t
x = vit + ½a(t)2
vf2 = vi2 + 2ax
Example: A jet plane lands with a velocity of ____ m/sec and can slow down (-acceleration) at a maximum
rate of –5.0 m/s2. Find (a) the time required for the plane to come to rest, and (b) the minimum size of
the runway.
(a)
(b)
Example: A train is traveling down a straight track at ___ m/sec when the engineer applies the brakes,
resulting in an acceleration of –1m/sec2 as long as the train is in motion. How far does the train travel in
the first ___ seconds after the breaks are applied?
Example: A racing car starting from rest accelerates at a rate of ____ m/s2. What is the velocity of
the car after it has traveled 100. ft?
FALLING OBJECTS
A body is said to be in ______________________ when the only force acting on it is gravity.
A body in free fall near the surface of the Earth will have a constant acceleration.
This acceleration due to gravity, g =
This implies that heavy objects fall at the same rate as light objects and in a vacuum they do.
Quick Quiz: An object is dropped from rest at t = 0s. It falls freely with constant acceleration of 9.8
m/s2, which implies that:
Quick Quiz: Ignoring air resistance, if you drop an object, it accelerates downward at 9.8m/s2. If instead
you throw it down, what will be its acceleration after you release it?
Quick Quiz: You throw a ball straight up. At its highest point, what are the magnitudes of ball’s velocity
and acceleration?
Example:
A ball is thrown upward from the top of a 35.0m building. The ball has an initial velocity of 6.5 m/s.
Diagram:
Note: we are still dealing with
motion in one dimension. The
diagram does not accurately
reflect the path of the ball, which
is actually moving straight up and
then straight back down.
A. Calculate how high the ball will rise.
B. Calculate how fast will the ball be going when it hits ground.
Example: An object is thrown vertically into the air with an initial velocity of 100 ft/s. At what time is it
at an elevation of 24 ft? (Careful with UNTIS!)
Free Fall Motion:
Free Fall Motion – Symmetry:
Chapter 3 Notes – Motion in Two Dimensions
Introduction to Vectors
Scalars and Vectors





A quantity is something that you measure.
SCALAR quantities have only __________, or ___________________.
VECTOR quantities have both a _________ and a __________________ (such as N, S, E or W).
Vectors are symbolized with arrows.
Adding vectors is like following a treasure map.
Example: Add the following vectors…
Now add the same vectors in a different order…
N
N
W
E
W
E
S
S
The vector sum __________________________________________________________________.
Vector Operations
A vector has components. If the components are on the axis they are called rectangular components.
The sum of a vector’s components equals the vector.
A vector and its components are interchangeable. You can either use the vector or its components,
depending on which is easier.
Finding Vector Components:
SOHCAHTOA
sin =
cos  =
tan  =
Now solve for the vector components:
NOTE: the x component is not always the cos function and the y component is not always the sin function.
For example:
Adding vectors in real life…
Step 1: draw a vector diagram
Step 2: Create data table holding x and y components of each vector and the total x and y components of
the resultant vector.
Step 3: add the vectors along each axis and put in data table
Step 4: draw a vector diagram showing only the vector axis sums from step 3
Step 5: use the Pythagorean Theorem (a2 + b2 = c2) to find the magnitude of the resultant vector.
Step 6a: Use a trig function (usually tan) to find the angle.
Step 6b: specify both magnitude and direction of the vector.
Example: Add the vectors below.
Projectile Motion
Projectile motion assumptions:
The acceleration of gravity is a constant _______________.
The effect of _____________________________ is negligible.
The ____________________ of the earth has no effect.
Projectile motion applies only to bodies in free fall.
The motion of the y-axis is ________________________ of the motion of the x-axis.
Motion on the y-axis is _______________________________________.
Motion on the x-axis is _______________________________________.
Without gravity, an object in motion will continue in motion with the same speed and in the same direction.
Horizontal displacement:
Vertical displacement:
This leads to a parabolic path.
Example #1: A cannon has a muzzle velocity of 62.3 m/s What is its range (_____ = ?) when shot at an
angle of 30.00?
1. Draw a vector diagram and resolve the velocity vector into rectangular components.
2. Using the y-axis component and the equations of motion for free fall, calculate the time of
flight. (Determine how long the projectile is in the air.)
3. Using the time of flight, calculate how far the projectile will travel horizontally during that
time.
The maximum range of a projectile occurs at _______.
Misconception #1: Going faster horizontally means you don’t fall as fast.
Misconception #2: Gravity won’t act on you until you look down. (think cartoons here)
Example #2: A golfer makes a shot to a tee as shown. Assuming he shoots at a 60.0 angle with a velocity
of 100. ft/sec, what is the range (R) to the tee? (remember g = 9.8 m/s2 or 32 ft/s2)
Relative Motion
Example 1: A passenger at the rear of a train, traveling at 15 m/s relative to the Earth, throws a
baseball with a speed of 15 m/s in the opposite direction to the motion of the train. What is the velocity
of the baseball relative to Earth?
Example 2: A boat heading due north crosses a wide river with a velocity of 10.0 km/hr relative to the
water. The river has a uniform velocity of 5.00 km/hr due east. Determine the velocity of the boat with
respect to an observer on the riverbank.
Example 3: A man on a boat walks at a rate of 2.0 m/s in a direction towards starboard (right of the
boat). The boat is sailing North at a rate of 8.0 m/s relative to the water. The water is flowing SW
relative to the Earth at a rate of 5.00 m/s. What is the man’s rate vector relative to the Earth?