Linear Motion or
One Dimensional Kinematics
SP1. Students will analyze the relationships between force, mass,
gravity, and the motion of objects.
a. Calculate average velocity, instantaneous velocity, and
acceleration in a given frame of reference.
b. Compare and contrast scalar and vector quantities.
c. Compare graphically and algebraically the relationships
among position, velocity, acceleration, and time.
Introduction
• Kinematics is the science of describing the
motion of objects using words, diagrams,
numbers, graphs, and equations.
• Kinematics is a branch of mechanics. The goal
of any study of kinematics is to develop
sophisticated mental models which serve to
describe (and ultimately, explain) the motion
of real-world objects.
Introduction
• Physics is a mathematical science. The
underlying concepts and principles have a
mathematical basis.
• Throughout the course of our study of physics,
we will encounter a variety of concepts which
have a mathematical basis associated with
them.
• It is imperative that you have a functional
scientific calculator with you from now on –
NOT A CELL PHONE CALCULATOR, as those
will NOT be permitted on tests/quizzes.
Scalars and Vectors
• Scalars are quantities which are fully described
by a magnitude alone.
– Magnitude refers to the size of a measurement
• Vectors are quantities which are fully described
by both a magnitude and a direction.
– Direction can be given as a movement through
space (N S E W) or within a coordinate plane (+ or -).
– Vectors are represented visually with arrows
pointing in the direction of movement.
Scalars and Vectors
Check Your Understanding:
Categorize each quantity as being either a vector
or a scalar.
QUANTITY
a. -5 cm
b. 256 bytes
c. 30 m/sec, East
d. 80 degrees Celsius
e. 4000 Calories
f. 7 km, 50° NW
CATEGORY
MATH ALERT!
• Because vectors involve direction, the
problems will often involve combining values
that either are or are not in the same
direction.
• Vectors are always connected head to tail,
and it does not matter in which order.
MATH ALERT!
• For single axis (all X or all Y) measurements,
we connect the vectors head to tail and
simply add the values together, retaining the
sign to represent direction.
MATH ALERT!
• For dual axis (both X and Y) measurements,
we connect the component vectors head to
tail to create a right triangle.
• Connect the initial to the final position with a
diagonal vector (i.e. the hypotenuse), called
the resultant.
• Use the Pythagorean theorem to calculate the
magnitude of the resultant vector.
• Use trig functions to calculate the direction of
the resultant vector.
MATH ALERT!
Distance and Displacement
• Distance is a scalar quantity which refers to
"how much ground an object has covered"
during its motion.
– It can never be negative, never decreases, and can
have multiple directions.
• Displacement is a vector quantity which refers
to "how far out of original position an object is“
– It is the object's overall change (Δ) in position Δ =
– Displacement can be negative, represents the final –
shortest path between two points, and has a initial
singular direction.
Distance and Displacement
Distance vs. Displacement
Check Your Understanding:
1. You walk 4 meters East, 2 meters South, 4
meters West, and finally 2 meters North.
A. What is the distance covered? __________ m
B. What is the displacement? __________ m _____
Distance vs. Displacement
Check Your Understanding:
2. You walk 3 meters East, 7 meters South, 7
meters West, and finally 7 meters North.
A. What is the distance covered? __________ m
B. What is the displacement? __________ m _____
Distance vs. Displacement
Check Your Understanding:
3. You walk 3 meters East, 4 meters South, 5
meters West, and finally 6 meters North.
A. What is the distance covered? __________ m
B. What is the displacement? __________ m _____
Speed and Velocity
• Speed is a scalar quantity which refers to "how
fast an object is moving.“
– Speed can be thought of as the rate at which an
object covers distance, or distance per time.
– A fast-moving object has a high speed; a slowmoving object has a low speed.
– An object with no movement at all has a zero speed.
Instantaneous Speed
and Average Speed
• Instantaneous speed is the speed at any given
instant in time.
– What the speedometer reads when you look at it
• Average speed is the average of all
instantaneous speeds
– The average of an infinite number of speedometer
readings during a trip
– Found simply by a total distance/total time ratio
Speed and Velocity
• Velocity is a vector quantity which refers to "the
rate at which an object changes its position.“
– Velocity can be thought of as the rate at which an
object covers displacement, or displacement per
time.
• Average velocity is calculated by a total
displacement/total time ratio, with
consideration of direction.
Average Speed vs. Average Velocity
Check Your Understanding:
1. While on vacation, you traveled from Augusta
to Atlanta for a total distance of 234 km. Your
trip took 2.5 hours. What was your average
speed? ________ km/hr
Can you determine average velocity?
Average Speed vs. Average Velocity
Check Your Understanding:
2. You walk 3 meters East, 7 meters South, 7
meters West, and finally 7 meters North. The
entire motion lasted for 24 seconds.
Average speed = _______ m/s
Average velocity = _________ m/s _____
Average Speed vs. Average Velocity
Check Your Understanding:
3. Joe is driving to Macon. The first leg of the trip
is via I-20 W, where he travels 144 km to
Madison. He then takes US 129 S for 70 km to
Macon. The trip takes 2.25 hr.
Average speed = _______ m/s
Average velocity = _________ m/s _____
More Problems
1. A runner makes one lap around a 200 m track in a time of
25.0 s. What was the runner's average speed?
2. If a car moves with an average speed of 60 km/hr, how
far will it travel if it continues this average rate for 4 hrs?
3. A bullet is shot from a rifle with a speed of 720 m/s.
What time is required for the bullet to strike a target
3240 m away?
More Problems
4. A bird is heading toward a feeder for 11 m
north. It then travels east for 8 m and lands on a
perch. The whole trip takes 25 s. What is the
bird’s average velocity?
Position – Time Graphs
• The slope of the line on a position versus time
graph is equal to the velocity of the object.
xf – xi
tf - ti
• Check Your Understanding:
Determine the velocity (i.e., slope) of
the object as portrayed by the graph.
Position – Time Graphs
Position – Time Graphs
Constant Velocity
Slow, Positive
Constant Velocity
Fast, Positive
Constant Velocity
Slow, Negative
Constant Velocity
Fast, Negative
Acceleration
• Acceleration is a vector quantity which is defined
as the rate at which an object changes its
velocity.
– An object is accelerating if it is changing its velocity.
• For objects with a constant acceleration, the
distance of travel is directly proportional to the
square of the time of travel.
Acceleration
Acceleration
• The direction of the acceleration vector depends
on two things:
whether the object is speeding up or slowing
down
whether the object is moving in the + or direction
• The general RULE OF THUMB is:
If an object is slowing down, then its
acceleration is in the opposite direction of its
motion.
Animation
Average Acceleration
Check Your Understanding:
1. Use the equation for acceleration to determine
the acceleration for the following two motions.
• Acceleration A = _________ m/s/s or m/s2
• Acceleration B = _________ m/s/s or m/s2
Average Acceleration
Check Your Understanding:
2. A roller coaster car rapidly picks up speed as it
rolls down a slope. As it starts down the slope,
its speed is 4 m/s. Three seconds later, at the
bottom of the slope, its speed is 22 m/s. What
is its average acceleration?
Average Acceleration
Check Your Understanding:
3. You are traveling in a car that is moving at a
velocity of 20 m/s. Suddenly, a car 10 meters in
front of you slams on its brakes. At that
moment, you also slam on your brakes and
slow to 5 m/s. Calculate the acceleration if it
took 2 seconds to slow your car down.
More Problems
4. Drag racers are able to accelerate at 12.5 m/s2 from
rest to 100 m/s before crossing the finish line. How
much time elapses during the run?
5. A bus decelerates at -8.3 m/s for 3.0 s to pick up
passengers. What is the bus’s velocity change?
6. You ride for 2 hours on cruise control at 70 mi/hr.
What is your average acceleration?
Kinematic Equations
v = Δx
t
a = Δv
t
vf = vi + at
vf2 = vi2 + 2aΔx
Δx = vit + ½at2
Δx – displacement (how far?)
t – time (how long?)
a – acceleration
vi – initial velocity
vf – final velocity
Kinematic Example
Ima Hurryin is approaching a stoplight moving
with a velocity of +30.0 m/s. The light turns
yellow, and Ima applies the brakes and skids to a
stop. If Ima's acceleration is -8.00 m/s2, then
determine the displacement of the car during
the skidding process.
Free Fall and Gravitational
Acceleration
• A free-falling object is one which is falling
under the sole influence of gravity. This
definition of free fall leads to two
important characteristics about a freefalling object:
Free-falling objects do not encounter air
resistance
All free-falling objects (on Earth) accelerate
downwards at a rate of -9.8 m/s2
This rate is commonly referred to as g
Free Fall and Gravitational
Acceleration
• All of the equations that work for
HORIZONTAL (x-direction)
motion ALSO work for VERTICAL
(y-direction) motion.
• Simply substitute g for a and Δy
for Δx
• Generally, free falling objects
start falling from rest, so it tends
to simplify the equations by
eliminating vi
The Big Misconception
• The acceleration of gravity, g, is the same for all freefalling objects regardless of how long they have been
falling, or whether they were initially dropped from
rest or thrown up into the air.
• BUT "Wouldn't an elephant free-fall faster than a
mouse?"
 NO!!
• WHY?
• All objects free fall at the same rate of acceleration,
regardless of their mass.
Position – Time Graphs
• Recall that the slope of the line on a position
versus time graph is equal to the velocity of the
object.
• This is really easy to calculate for objects
moving at a constant velocity (i.e. having NO
acceleration)
Position – Time Graphs
• However, for objects WITH accelerated motion,
the graphs look quite different:
Changing Velocity
Positive
Changing Velocity
Slow to Fast, Negative
Changing Velocity
Fast to Slow, Negative
Acceleration Graphs
• Calculating the slope of those curved lines
requires calculus, so we won’t be doing
that…so, how do we determine the value of
acceleration from a graph?
• We use VELOCITY-TIME graphs!
• The slope of the line on a velocity-time graph is
acceleration:
vf – v i
tf - t i
Velocity – Time Graphs
Positive Velocity
Zero Acceleration
Velocity – Time Graphs
Check Your Understanding
The velocity-time graph for a two-stage rocket is shown
below. Use the graph and your understanding of slope
calculations to determine the acceleration of the rocket
during the identified time intervals.
Ticker Tape Diagrams
• A common way of analyzing the motion of
objects in physics labs is to perform a ticker tape
analysis.
• A long paper tape is attached to a moving object
and threaded through a device that places a
mark on the tape at regular intervals of time say every 0.10 second.
• As the object moves, it drags the tape through
the "ticker," thus leaving a trail of dots. The trail
of dots provides a history of the object's motion
and therefore a representation of the object's
motion.
Ticker Tape Diagrams
Ticker Tape Diagrams
Check Your Understanding:
Imagine a car with a leaky engine that drips oil at a regular rate.
As the car travels through town, it would leave a trace of oil on
the street that would reveal information about the motion of
the car.
Analyze the three traces of Renatta Oyle's car as shown below.
1.
2.
3.