One Dimensional Motion
Honors Physics
Fall, 2014
Vector quantities
• Anything with
MAGNITUDE and
DIRECTION is
termed a vector
quantity
– Scalar quantities just
have magnitude
Gravity propels a skier down
a snow-covered slope at an
acceleration approximately
constant. The equations of
“KINEMATICS”, as studied in
this chapter, can give his
position and velocity at any
given time
Position, Distance, and
Displacement
• Coordinate system  defines position
• Distance  total length of travel
– (SI unit = meter, m)
– Scalar quantity
• Displacement  change in position
Position, Distance, and
Displacement
Before describing motion, you
must set up a coordinate system –
define an origin and a positive
direction.
The distance is the total
length of travel; if you
drive from your house to
the grocery store and
back, what is the total
distance you traveled?
Displacement is the change in position. If you drive from your house to
the grocery store and then to your friend’s house, what is your total
distance?
What is your displacement?
Average speed and velocity
• Average speed = distance / time
• Average velocity  displacement
divided by the total elapsed time
displacement
Averagevelocity
elapsed time
 x x f  xi
vav 

t
t f  ti
Average speed and velocity
What’s his average
velocity if he returns
to his starting point?
What is his average
velocity if he sprints
50 m in 8 s?
What’s his average
velocity if he walks
back to the starting
line in 40 s?
Displacement
Time taken
Displacement
and Velocity
in One
Dimension
Graphical Interpretation of Velocity
• The left graph shows a car moving at constant
velocity (linear). The graph on the right shows a
car with changing velocity. The average velocity
for a given time interval is the slope of the line
connecting the two coordinates in question.
Graphical Interpretation of Average
Velocity
• The same motion,
plotted onedimensionally and as
an position vs. time (x-t)
graph:
Position vs time graphs give us information about:
• average velocity  slope of a line on a x-t plot is
equal to the average velocity over that interval
rise y2  y1
slope 

run x2  x1
Graphical Interpretation of
average velocity
What’s the average velocity between the intervals t = 0 s  t = 3 s? Is the
velocity positive or negative?
What’s the average velocity between the intervals t = 2 s  t = 3 s? Is the
velocity positive or negative?
• Which object is
moving with
constant
POSITIVE
velocity? Which
is moving with
NEGATIVE
velocity? Which
isn’t moving?
• HOW DO YOU
KNOW?
Instantaneous Velocity
• Instantaneous velocity
• This means that we evaluate the average
velocity over a shorter and shorter period
of time
Instantaneous velocity
• If we have a more complex motion…
• This plot shows
the average velocity
being measured
over shorter and
shorter intervals.
The instantaneous
velocity is the
SLOPE OF THE
tangent
LINE
to the curve.
Instantaneous velocity
Average velocity is the slope of the straight line connecting two points corresponding to
a given time interval
Instantaneous velocity is the slope of the tangent line at a given instant of time
Instantaneous velocity
Is the
Instantaneous
velocity at t = 0.5 s
A.Greater than
B.Less than
C.Or equal to
the instantaneous
velocity at t =1.0 s?
How do you know?
Displacement
and Velocity in
One Dimension
Are the plots shown at the left
correctly related
A) YES
B) NO
CAN YOU EXPLAIN
WHY?!?!
Mechanics Lecture 1,
Slide 18
THERE’S ROOM
OVER THERE 
YOU KNOW…
The velocity vs. time plot of
some object is shown to the
right.
Which diagram below could
be the Displacement vs.
time plot for the same
object?
A
B
C
Acceleration
• Average acceleration  the change in
velocity divided by the time it took to
change the velocity
Acceleration
• On Earth, gravitational
acceleration equals about
10 m/s/s
• What does it mean to
have an acceleration of
10 m/s2 ?
Time (s)
Velocity (m/s)
0
0
1
10
2
20
3
30
Graphical interpretation of
Acceleration
Important Point Regarding
Acceleration
• When an object’s velocity and acceleration (both
vector quantities) occur in the same direction,
the object is SPEEDING UP!!!!
• When an object’s velocity and acceleration occur
in opposing directions, the object is
SLOWING DOWN!!!
– Deceleration is used to refer to a decrease in speed
– Don’t confuse “negative acceleration” with
deceleration…PLEASE!!
Instantaneous Acceleration
• Very similar to
“instantaneous velocity”
from our position vs.
time graphs
• a = lim Δv / Δt
t0
• The closer Δt gets to
zero, the closer our ratio
gets to a fixed number.
Acceleration
• Velocity vs time (v-t) graphs give us information
about: average acceleration, instantaneous
acceleration
• the “+” 0.25 m/s2 means the particle’s speed is
increasing by 0.25 m/s every second
For the Displacement and Velocity curves
shown on the left, which is the correct plot of
acceleration vs. time?
A
B
1-D Motion with constant
acceleration
• When an object moves at
constant acceleration, the
instantaneous acceleration at
any point in a time interval
equals the average
acceleration divided by the
whole time interval
• In other words, with constant
acceleration:
– Average acceleration is no
different than instantaneous
acceleration!
Motion with constant
acceleration
• If the acceleration is constant, the velocity
changes linearly:
•
•
•
•
Average velocity:
v0 = initial velocity
a = acceleration
t = time
Constant Acceleration
constant
a(t) = a
Finding Displacement on a graph
not using the origin as initial
coordinates
• What this graph
shows is the
geometrical
interpretation of
the following
equation! Cool,
huh?!
Freely Falling Objects
• All objects fall
towards earth at the
same constant
acceleration
• Assuming air resistance
is zero, of course!!
• Any object moving
upward, downward, or
released from rest
under the influence of
gravity is considered
“free falling”
• A football in the air,
skydiver, falling book,
etc.
Motion with constant acceleration
An object falling in air is subject to air
resistance (and therefore is not freely falling).
• Free fall is the motion of an object subject only to
the influences of gravity…in most cases we’ll
consider, air resistance is negligible and can be
ignored.
Free falling
objects
Free fall from rest:
You could use the
kinematics equations
to see where the
numbers are coming
from, fyi!!!
Trajectory of a projectile