1D Motion
Physics 2015-2016
Vector and Scalar Quantities
• Vector quantities need both magnitude (size)
and direction to completely describe them
– Ex: Displacement, Velocity, Acceleration
• Scalar quantities are completely described by
magnitude only
– Ex: Distance travelled, speed, Energy, time
Position
x  x f  xi
Frame of Reference Practice
• Describe your position in three different
reference frames.
Displacement
Displacement Isn’t Distance
• The displacement of an object is not the same
as the distance it travels
– Ex: Throw a ball straight up and then catch it at
the same point you released it
• The distance is twice the height – 2h
• The displacement is zero
h
Position, Displacement and Distance
travelled – Practice WB
State the position of the car at:
A: ____
E: ____
State the distance travelled
from:
A to B: _____
A to D: _____
A to F: ______
State the displacement of the
car from:
A to B: _____
A to D: _____
A to F: _____
Write:
• What is the difference between position,
distance travelled and displacement.
Average Speed
• The average speed of an object is defined as the
total distance traveled divided by the total time
elapsed
total distance
Average speed 
total time
• Scalar quantity
• Ignores any variations in the object’s actual motion during the
trip
• The total distance and the total time are all that is important
• SI units = m/s
Average Velocity
• The average velocity is rate at which the
displacement occurs
v average
xf  xi
x


t
tf  ti
• Direction will be the same as the direction of the
displacement (time interval is always positive)
–
+ or - is sufficient for now
• SI Units of velocity are m/s
– (may need to be converted)
– Ignores actual movement as well
Average Speed and Velocity Practice - wb
State the Average Speed if it
takes 20 seconds to travel :
From A to B: ______
From A to D: ______
From A to F: _______
From F to A: _______
State the Average Velocity if it
takes 20 seconds to travel:
From A to B: _______
From A to D: _______
From A to F: ________
From F to A:
Speed vs. Velocity
 Cars on both paths have the same average velocity since
they had the same displacement in the same time
interval
 The car on the blue path will have a greater average
speed since the distance it traveled is larger
Practice #1 - WB
An athlete swims from the north end to the
south end of a 50.0 m pool in 20.0 seconds
and makes the return trip to the starting
position in 22.0 s.
› What is the average velocity for the first half of
the swim? ________
› What is the average velocity of the second half of
the swim? _______
› What is the average velocity for the round trip?
_______
Practice #2
• Example: Two friends bicycle 3.0 kilometers
north and then turn to bike 4.0 kilometers
east in 25 minutes.
• a) What is their average speed?
• b) What is their average velocity?
Uniform Motion/Constant Velocity
• Uniform motion = constant velocity : a =0
• The displacement changes by equal amounts
in equal intervals of time
Time (s)
Distance (m)
Velocity (m/s)
Acceleration (m/s2)
0
1
2
3
4
HP: Uniform Motion Formula
• Displacement at t = 0 is xo
Displacement at time t is x
t=0
t=t
xo
And, v   x  x  x o
t
t 0
Then:
x = xo+vt
x
HP: Uniform Motion Examples
1.
a. A car initially at position -5.0 m has a uniform
velocity of 7.0 m/s.
– At what position is it 4.0 seconds later?
b. A car initially at position 4 m has a velocity of
-3 m/s.
– At what time does it reach a position of -20
meters?
– What distance does the car travel in this time?
Physics 500 Lab
• Regular Physics only
HP - Instantaneous Velocity
• The limit of the average velocity as the time interval
becomes infinitesimally short, or as the time interval
approaches zero
x
x
v 
or v=
t
t
• The instantaneous velocity indicates what the
velocity is at every point of time, not just over a
period of time
• Example: Car is speeding up and slowing down –
average velocity doesn’t tell us much about its
movement.
lim
t 0
Position vs. Time Graphs
• Computer/Vernier in groups
s/m
Graphical Interpretation:
distance vs. time - wb
•Not Moving
•Constant velocity
•Constant velocity of
5ms-1
•Negative velocity of
5ms-1
•Starting from rest and
accelerating
•Starting 20 meters
away at a fast pace and
slowing down.
t/s
Graphical Interpretation of Velocity
• Velocity can be determined from a positiontime graph
• Average velocity equals the slope of the line
joining the initial and final positions of a
distance – time graph. UNITS can prove it!
• An object moving with a constant velocity will
have a graph that is a straight line
*Average Velocity, Constant
• The straight line
indicates constant
velocity
• The slope of the line is
the value of the
average velocity
Average Velocity, Non Constant
 The motion is nonconstant velocity
 The average velocity
is the slope of the
blue line joining two
points
HP - Instantaneous Velocity on a
Graph
• The slope of the line tangent
to the position-vs.-time
graph is defined to be the
instantaneous velocity at
that time
– The instantaneous speed
is defined as the
magnitude of the
instantaneous velocity
100m Split times - discussion
Acceleration
vf  vi
a
t
or
v
a
t
Now Try Practice B
Velocity vs. Time graphs
• Computer/Vernier in groups
Graphical Interpretation: velocity vs.
time
v/ms-1
•
•
•
•
t/s
Not moving
Constant velocity
Accelerating
Slowing down
HP - Using Velocity vs. time graphs
to calculate displacement.
•The area below a velocity – time graph represents the total
displacement between those two times.
• Prove it with UNITS!
•Example:
HP - Graphical Interpretation of
Acceleration
• Average acceleration is the slope of the line
connecting the initial and final velocities on a
velocity-time graph – UNITS can prove it!
• HP - Instantaneous acceleration is the slope of
the tangent to the curve of the velocity-time
graph
Average Acceleration
Uniformly Accelerated Motion
•Uniformly Accelerated Motion = Constant
Acceleration
•The velocity changes equal amounts in equal
amounts of time.
Time (s)
0
1
2
3
Distance (m)
0
3
10
23
Velocity (m/s)
0
5
10
15
Acceleration (m/s2)
5
5
5
5
HP - Some practice
For each simulation, state whether each is
positive, negative or zero.
Displacement
Velocity
Acceleration
HP - Some Practice
Quiz #1
• Definitions and finding distance,
displacement, speed, velocity, acceleration
• Calculations of average velocity (Practice A)
and average acceleration (Practice B)
• Interpreting, creating and finding values from
position vs. time and velocity vs. time graphs
Problem-Solving Requirements
• Read the problem
• Draw a diagram!!
– Choose a coordinate system, label initial and final points, indicate a
positive direction for velocities and accelerations
• Label all known quantities, be sure all the units are consistent
– Convert if necessary
• Choose the appropriate kinematic equation(s) that has only one variable
that you don’t have the value for.
• Solve for the unknowns
– You may have to solve two equations for two unknowns. Be careful of
Negatives!
• Check your results
– Estimate and compare
– Check units
Constant Acceleration Formula
v v f  v i
a

t
t 0
2. A plane starting at rest at one end of a
runway undergoes a uniform acceleration of
4.8 m/s2 for 15 s before takeoff. What is its
speed at takeoff? How long must the runway
be for the plane to be able to take off?
Constant Acceleration Example
3. An object starting with an initial velocity of
5.0 m/s undergoes constant acceleration. After
10 seconds its velocity is found to be 35 m/s.
– What is the acceleration?
– Is it possible to know its position?
4. An object starting with an initial velocity of
2 m/s undergoes a constant acceleration of 4
m/s2.
– When does it reach a velocity of 14 m/s?
Now Try: Practice D
Another Constant Acceleration
Formula
Example
5. A racing car reaches a speed of 42 m/s. It then
begins a uniform negative
acceleration, using its parachute and braking
system, and comes to rest
5.5 s later. Find the distance that the car travels
during braking.
Now Try: Practice C
Another Constant Acceleration
Formula
6. A person pushing a stroller starts from rest,
uniformly accelerating at a rate of 0.500 m/s2.
What is the velocity of the stroller after it has
traveled 4.75 m?
• Now Try Practice E
Relationship Between
Acceleration and Velocity
• Uniform velocity (shown by red arrows
maintaining the same size)
• Acceleration equals zero
Relationship Between Velocity and
Acceleration
 Velocity and acceleration are in the same direction
 Acceleration is uniform (blue arrows maintain the same
length)
 Velocity is increasing (red arrows are getting longer)
 Positive velocity and positive acceleration
Relationship Between Velocity and
Acceleration
 Acceleration and velocity are in opposite directions
 Acceleration is uniform (blue arrows maintain the same
length)
 Velocity is decreasing (red arrows are getting shorter)
 Velocity is positive and acceleration is negative
HP - Uniformly Accelerated Motion Equations
• Used in situations with uniform/constant
acceleration and uniform/constant
acceleration ONLY!!!!!
From Definitions:
Derived Formulas:
s  v ave  t
vi

x 
v f  v i  at
v ave
vf  vi

2
 vf 
2
t
1 2
x  v i t  at
2
2
2
v f  v i  2a(x )
Deriving the equations
HP - Examples:
• 7. Nathan accelerates his skateboard
uniformly along a straight path from rest to
12.5 m/s in 2.5 sec.
A. What is Nathan’s displacement during
this time interval?
B. What is Nathan’s average velocity during
this time interval?
HP - Example:
8. A body at the origin has an initial velocity of 6.0 ms-1 and moves with an acceleration of 2.0
ms-2 . When will its displacement become 16
m?
HP - Example:
9. A car starts from rest and travels for 5 seconds
with a uniform acceleration of +1.5m/s2. The
driver then applies the brakes, causing a uniform
acceleration of -2 m/s2. If the brakes are applied
for 3 seconds, how fast is the car going at the
end of the braking period, and how far has it
gone from its start?
Frames of Reference
• Video
• We are used to measuring velocities with
respect to observers who are “at rest”
however, if you are moving, you are in your
own frame of reference and observe things
accordingly.
Relative Motion and Frames of
Reference
• You are waiting for a bus
• A bus passes you at 20 m/s
• A passenger is walking on the bus at a speed
of 1 m/s
– What is his relative velocity if he walks in the
direction the bus is going
– What is his relative velocity if he walks in the
direction opposite the bus.
Relative Motion and Frames of
Reference Example
1. Car 1 – 35 km/hr
Car 2 – 40 km/hr
Car 2 overtakes Car 1 because it is going faster
What is the relative speed of Car 1 to Car 2
Free Fall and Air resistance
Galileo Galilei
• 1564 - 1642
• Galileo formulated the
laws that govern the
motion of objects in free
fall
• Also looked at:
–
–
–
–
Inclined planes
Relative motion
Thermometers
Pendulum
Free Fall
• Video
• Galileo’s thought experiment
• All objects moving under the influence of gravity only
are said to be in free fall
– Free fall does not depend on the object’s original motion
• All objects falling near the earth’s surface fall with a
constant acceleration
• The acceleration is called the acceleration due to
gravity, and indicated by g
Acceleration due to Gravity
• Symbolized by g
• g = 9.8 m/s² or 9.81 m/s2
– When estimating, use g 10 m/s2
• g is always directed downward
– toward the center of the earth – typically negative in the
equations
• Ignoring air resistance and assuming g doesn’t vary
with altitude over short vertical distances, free fall is
constantly accelerated motion
Free Fall – an object dropped
• Initial velocity is zero
• Let up be positive
• Use the kinematic
equations
– Generally use y instead
of x since vertical
• Acceleration is
g = -9.80 m/s2
U=0
a=g
Free Fall – an object thrown
downward
• a = g = -9.80 m/s2
• Initial velocity  0
– With upward being
positive, initial velocity
will be negative
Free Fall -- object thrown upward
• Initial velocity is positive
• The instantaneous
velocity at the maximum
height is zero
• a = g = -9.80 m/s2
everywhere in the motion
v=0
Thrown upward, cont.
• The motion may be symmetrical
– Then tup = tdown
– Then v = -u
• The motion may not be symmetrical
– Break the motion into various parts
• Generally up and down
Non-symmetrical
Free Fall
• Need to divide the motion
into segments
• Possibilities include
– Upward and downward
portions
– The symmetrical portion
back to the release point
and then the nonsymmetrical portion
Free fall Example #1
• A pebble is dropped down a well and hits the
water 1.5 seconds later. Using the equations
for motion with constant acceleration,
– Determine the distance from the edge of the well
to the water’s surface.
– For the same well, determine how long it would
take to hit the water if it was thrown down with a
speed of 1 m/s.
Free Fall Example #2
• A person throws a ball upward into the air
with an initial velocity of 15.0 m/s. Calculate:
– How high it goes
– How long the ball is in the air before it comes back
to his hand
– How much later the ball hits the ground if he
didn’t catch it, if his hand is 1 meters above the
ground.
Free Fall Example #3
• A mass is thrown upwards with an initial
velocity of 30 m/s. A second mass is dropped
from directly above, a height of 60 m from the
first mass, 0.5 s later.
– When do the masses meet?
– How high is the point where they meet?
Falling Cats
• Study was done to find out how cats survived
long falls.
• Results: The higher the cats were dropped
from, the less injuries occurred.
• WHY????
Investigating Air resistance
• Take a few minutes to experiment with dropping some objects
and thinking about your everyday experiences.
Forces and Acceleration
• Think of the times when you are accelerating:
in the car speeding up, in the car slowing
down, on a roller coaster, skiing off a jump.
• Now think of the times when you are not
accelerating – travelling at a constant speed in
the car, sitting at rest at your desk.
• How do you feel? If your eyes are closed, can
you feel whether you are accelerating or not?
Air Resistance and Terminal Velocity
T=0
V =0
T= 1
V = 10m/s
T =2
V=20m/s
T =3
V=30m/s