Displacement, Velocity, and
Acceleration
(WHERE
What we’re concerning ourselves with today is determining where and when in movement. How do we find the motion of stuff? We’re going to do it in a precise and rigorous way.
We will start with the simplified case of only moving along one axis of movement.
and
WHEN?)
Math resources
• Appendix A in your book!
• Symbols and meaning
• Algebra
• Geometry (volumes, etc.)
• Trigonometry
• Logarithms
In case any of you are struggling with remembering the math or a particular math concept…
Appendix A
Reminder
• You will do well in this class by PRACTICING!
Extra Practice Problems:
2.1, 2.3, 2.5, 2.21, 2.25, 2.27
Also: (Ungraded) homework warm-up problems
Also maybe 2.29, 2.31, 2.33
Reminders
Next class is next Wednesday.
Problem solving day: practicing for exam.
First clicker grade counted;
BRING YOUR CLICKERS!
Also maybe 2.29, 2.31, 2.33
Problem Solving Pro-tips
1. Draw a picture!
2. Use and label your reference frame.
3. List what you KNOW and DON’T KNOW in
variable form.
4. Practice helps you pick best formulas!
One of the MOST IMPORTANT PRACTICAL THINGS today will be some problem-solving tips.
Tip number 3 will help you pick the right formula to use!
Although tests will give you the necessary formulae, you will still need to pick the right one to use.
Scalars and Vectors
• Scalar: just a number (magnitude).
• Vector: a number (magnitude) with a direction.
In a lot of the work we’ll do in this class and anyone going into any science, we’ll need to know the difference between a scalar and a vector. These are just official-sounding terms for a simple concept. Here
are the definitions. “Direction can be east, up, positive, negative.”
[explain the graphic as it plays out]
Describing “Displacement” and “Distance” are something we’ll cover today.
Distance: total travelled. Displacement (in this case): Where it ended up IN REFERENCE TO where it started.
Vector: note that “+” is added, and it describes only the total displacement.
Scalars and Vectors
• Scalar: just a number (magnitude).
• Vector: a number (magnitude) with a direction.
30 m
100 m
In a lot of the work we’ll do in this class and anyone going into any science, we’ll need to know the difference between a scalar and a vector. These are just official-sounding terms for a simple concept. Here
are the definitions. “Direction can be east, up, positive, negative.”
[explain the graphic as it plays out]
Describing “Displacement” and “Distance” are something we’ll cover today.
Distance: total travelled. Displacement (in this case): Where it ended up IN REFERENCE TO where it started.
Vector: note that “+” is added, and it describes only the total displacement.
Scalars and Vectors
• Scalar: just a number (magnitude).
• Vector: a number (magnitude) with a direction.
30 m
100 m
Distance (scalar): 100m + 30m = 130 meters
In a lot of the work we’ll do in this class and anyone going into any science, we’ll need to know the difference between a scalar and a vector. These are just official-sounding terms for a simple concept. Here
are the definitions. “Direction can be east, up, positive, negative.”
[explain the graphic as it plays out]
Describing “Displacement” and “Distance” are something we’ll cover today.
Distance: total travelled. Displacement (in this case): Where it ended up IN REFERENCE TO where it started.
Vector: note that “+” is added, and it describes only the total displacement.
Scalars and Vectors
• Scalar: just a number (magnitude).
• Vector: a number (magnitude) with a direction.
Displacement, x (vector): 100 - 30 = +70 meters
Initial location
Final location
30 m
100 m
Distance (scalar): 100m + 30m = 130 meters
In a lot of the work we’ll do in this class and anyone going into any science, we’ll need to know the difference between a scalar and a vector. These are just official-sounding terms for a simple concept. Here
are the definitions. “Direction can be east, up, positive, negative.”
[explain the graphic as it plays out]
Describing “Displacement” and “Distance” are something we’ll cover today.
Distance: total travelled. Displacement (in this case): Where it ended up IN REFERENCE TO where it started.
Vector: note that “+” is added, and it describes only the total displacement.
Scalars and Vectors
Scalars:
Vectors:
Distance, x
Speed, v
Displacement, x
Velocity, v
Acceleration, a
Vectors are usually represented as BOLD
(or with an arrow hat).
DRAW ARROW HAT ON LIGHT BOARD.
This notation is a formality I don’t recommend stressing about until a few weeks from now when we start doing graphing in 2 dimensions.
Frames of reference
80 km/h
+10 km/h
70 km/h
Ground’s reference
frame
Velocity,
v
Driver’s reference
frame
• In ground frame of reference, one car has v = +80 km/h
while the other has v = +70 km/h
• In reference frame of driver, velocity of other car is
v = +10 km/h
I alluded to “reference frames” before, in that displacement was “destination in reference to starting point”. To note any vector, we need to define a reference point.
An intuitive way of understanding this is cars on highway…
Reference frames on paper
• PT #1: Draw a picture!
“Jogger went 10m east, 10m
north, sat on a stump a while,
then walked 25m east.”
First pro tip!
If you see a word problem that can be drawn, draw it.
Reference frames on paper
• PT #1: Draw a picture!
“Jogger went 10m east, 10m
north, sat on a stump a while,
then walked 25m east.”
First pro tip!
If you see a word problem that can be drawn, draw it.
Reference frames on paper
• PT #1: Draw a picture!
• PT #2: Use (and LABEL) a coordinate system.
+y
“Jogger went 10m east, 10m
north, sat on a stump a while,
then walked 25m east.”
-x
+x
0
-y
First two pro tips!
DRAW A REFERENCE FRAME (coordinate system)
The reference point here is the jogger’s starting point.
When you draw axes, point arrows in the positive direction.
[read the slide]
Will see an example on the next slide.
Reference frames on paper
• PT #1: Draw a picture!
• PT #2: Use (and LABEL) a coordinate system.
+y
“Jogger went 10m east, 10m
north, sat on a stump a while,
then walked 25m east.”
-x
+x
0
-y
The direction of these arrows is important for
setting up problems and may affect the sign of
your variables and/or answers (will see example soon)
First two pro tips!
DRAW A REFERENCE FRAME (coordinate system)
The reference point here is the jogger’s starting point.
When you draw axes, point arrows in the positive direction.
[read the slide]
Will see an example on the next slide.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Δx = x f − xi
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
This one’s easy, but
let’s practice pro tips!
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
This one’s easy, but
let’s practice pro tips!
Final position
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Initial position
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
This one’s easy, but
let’s practice pro tips!
Final position
Initial position
+x
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
This one’s easy, but
let’s practice pro tips!
Final position
Pro Tip #3: List what
you know & don’t
known in variable form
Initial position
+x
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
This one’s easy, but
let’s practice pro tips!
Final position
Pro Tip #3: List what
you know & don’t
known in variable form
Initial position
xi = +3.0 m
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
+x
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
This one’s easy, but
let’s practice pro tips!
Final position
xf = -5.0 m
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Pro Tip #3: List what
you know & don’t
known in variable form
Initial position
xi = +3.0 m
+x
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
This one’s easy, but
let’s practice pro tips!
Final position
xf = -5.0 m
Pro Tip #3: List what
you know & don’t
known in variable form
Initial position
xi = +3.0 m
Δx = ?
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical ref location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
+x
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
Final position
xf = -5.0 m
Initial position
xi = +3.0 m
Δx = -5.0 m - (+3.0 m) = -8.0 m
Definition of displacement is the change in position.
DELTA REPRESENTS CHANGE. HERE, Final minus initial position.
WALK THEM THROUGH THIS PROBLEM. The house seems to be the critical reference location here.
Pro tip… we know xi, we know xf.
WHAT DON’T WE KNOW?
+x
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
Final position
Δx = -8.0 m
Initial position
+x
Arrow represents the Δx vector:
magnitude (8.0m) and direction (-) of displacement.
Blue arrow from xi to xf indicates direction and magnitude of displacement - good example of a vector
Sign of displacement indicates direction - ALTHOUGH CAR DROVE OFF TO RIGHT, VECTOR POINTS TOWARD NEGATIVE.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
+y
Final position
Initial position
+x
Now I’m going to give you a conceptual test here, and point out the direction of the arrows for “positive” on this axis.
Now going to do something totally evil…
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
Initial position
Final position
+x
+y
Write your knowns and unknowns!
Trial problem: switch direction of positive.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
Initial position
Final position
+x
+y
Write your knowns and unknowns!
Trial problem: switch direction of positive.
Displacement (vector)
Definition: change in the position of an object
Displacement:
Δx = x f − xi
Ex: Car initially parked 3.0 m to right of house, drives around the
block, ends up 5.0 m to left of house. Find the displacement of the car.
Initial position
Final position
+x
xf = +5.0 m
+y
xi = -3.0 m
Δx = +5.0 m - (-3.0 m) = +8.0 m
Trial problem: switch direction of positive.
Many people struggle with signs! Ask
yourself after defining each variable:
Is the sign consistent with what
direction I’ve called positive?
Up and right are usually positive!
(particularly in WebAssign unless
explicitly stated in the problem)
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
SI units: m/s
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity going
to Pitt:
Mo’town
0
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Pitt
70 mi x
t = 2 hrs
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity going
to Pitt:
xi = 0
ti = 0
Mo’town
0
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Pitt
70 mi x
t = 2 hrs
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity going
to Pitt:
xi = 0
ti = 0
xf = +70 mi tf = 2 hrs
Mo’town
0
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Pitt
70 mi x
t = 2 hrs
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity going
to Pitt:
xi = 0
ti = 0
xf = +70 mi tf = 2 hrs
Mo’town
0
Pitt
70 mi x
t = 2 hrs
70 mi − 0
v=
= +35 mi/hr
2 hrs − 0
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity going
to Pitt:
xi = 0
ti = 0
xf = +70 mi tf = 2 hrs
Mo’town
0
Pitt
70 mi x
t = 2 hrs
70 mi − 0
v=
= +35 mi/hr
2 hrs − 0
Formally the definition is…
Another way to state this is “Change in LOCATION over period of TIME.”
Bar over symbol means average.
Most familiar miles per hour, kilometers per hour, but just to remind you of formal units, SI units is meters per second
Simple example…Over the course of 5 hours you go to pitt & back. Take 2hrs to go, and one hour to return.
Will walk you through this one and then give you one to chew on yourself.
DRAWN A PICTURE/AXES…
TIME ZERO IS REFERENCE POINT.
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity coming back from Pitt?
Average velocity of round trip?
If you finish those:
Average speed (scalar!) of round trip?
IF <13:00, HAVE THEM CALCULATE - group thought experiment.
OTHERWISE DO ON LIGHT BOARD.
WRITE ANSWERS ON LIGHT BOARD.
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity coming
back from Pitt:
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity coming
back from Pitt:
v=
0 − 70 mi
= −70 mi/hr
3 hrs − 2 hrs
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Average velocity coming
back from Pitt:
v=
0 − 70 mi
= −70 mi/hr
3 hrs − 2 hrs
Average velocity of round
trip:
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Pitt
Mo’town
0
70 mi x
Average velocity coming
back from Pitt:
v=
0 − 70 mi
= −70 mi/hr
3 hrs − 2 hrs
Average velocity of round
trip:
v=
0−0
=0
3 hrs − 0
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Average Velocity
Definition: velocity is displacement per unit time
Δx x f − xi
v≡
=
Δt t f − ti
SI units: m/s
Ex: Go to Pittsburgh in 2 hrs, back in Morgantown 3 hrs after leaving
Pitt
Mo’town
0
70 mi x
Average velocity coming
back from Pitt:
v=
0 − 70 mi
= −70 mi/hr
3 hrs − 2 hrs
Average velocity of round
trip:
v=
0−0
=0
3 hrs − 0
Speed: 140mi / 3h = 47 mi / h!
Important points here:
Identify START and END points for each problem.
Calculate
Note SIGN of return velocity!
Velocity is a VECTOR — sign indicates negative or positive direction!
Instantaneous Velocity
• Instantaneous velocity is velocity at a
particular instant.
• Only use the average velocity when asked for
“average.”
For example, I carefully planned my trip back from Pittsburgh so I would have an average speed home of 70mph (maybe because I had to be somewhere in an hour), but I got caught in traffic so I sped up. Does the cop
care if my average velocity is below the speed limit, but I was going 120mph at the point she caught me? No.
Instantaneous Velocity
• Instantaneous velocity is velocity at a
particular instant.
• Only use the average velocity when asked for
“average.”
For example, I carefully planned my trip back from Pittsburgh so I would have an average speed home of 70mph (maybe because I had to be somewhere in an hour), but I got caught in traffic so I sped up. Does the cop
care if my average velocity is below the speed limit, but I was going 120mph at the point she caught me? No.
Instantaneous Velocity
• Instantaneous velocity is velocity at a
particular instant.
• Only use the average velocity when asked for
“average.”
Will discuss this difference more next lecture.
For example, I carefully planned my trip back from Pittsburgh so I would have an average speed home of 70mph (maybe because I had to be somewhere in an hour), but I got caught in traffic so I sped up. Does the cop
care if my average velocity is below the speed limit, but I was going 120mph at the point she caught me? No.
Acceleration
• Average acceleration = change in velocity/time
v f − vi
Δv
a≡
=
t f − ti Δt
• Instantaneous acceleration
Δv
a = lim
Δt →0 Δt
SI Units:
m/s/s = m/s2
Sign of acceleration definitely harder for people to visualize than sign of displacement or velocity. If time, maybe go through example with desk as origin and me moving around to talk about sign of position, velocity
and acceleration.
Acceleration
• Average acceleration = change in velocity/time
v f − vi
Δv
a≡
=
t f − ti Δt
• Instantaneous acceleration
Δv
a = lim
Δt →0 Δt
SI Units:
m/s/s = m/s2
The sign of acceleration indicates which direction its velocity changes.
Positive acceleration means speeding up when moving in the positive x
direction OR slowing down when moving in the negative x direction.
Sign of acceleration definitely harder for people to visualize than sign of displacement or velocity. If time, maybe go through example with desk as origin and me moving around to talk about sign of position, velocity
and acceleration.
Signs of acceleration
• A car slowing down at a stop sign
a
v
+x
• A bullet hitting a wall
v
a
+x
• Sprinter out of the blocks
av
+x
The sign of acceleration indicates which direction its velocity changes. Positive acceleration means speeding up when moving in the positive x direction OR slowing
down when moving in the negative x direction.
Drawing a picture and the v and a vectors will help!
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
“t at time zero”
“location at time zero”
“velocity at time zero”
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
“t at time zero”
“location at time zero”
“velocity at time zero”
v f − vi
t f − ti
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
v f − vi
t f − ti
a=
“t at time zero”
“location at time zero”
“velocity at time zero”
v − vo
t
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
v f − vi
t f − ti
a=
v − vo
t
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
a=
v − vo
t
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x f − xi
t f − ti
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
t f − ti
a=
v − vo
t
vavg =
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
t
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
t f − ti
a=
v − vo
t
vavg =
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
t
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
x = xo + vavg t
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
t f − ti
vavg =
a=
v − vo
t
vavg =
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
t
v + vo
2
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
x = xo + vavg t
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
a=
v − vo
t
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
x = xo + vavg t
t f − ti
t
Similar derivations lead to more equations:
vavg =
vavg =
v + vo
2
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
a=
v − vo
t
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
x = xo + vavg t
t f − ti
t
Similar derivations lead to more equations:
vavg =
v + vo
2
vavg =
Δx = vot + 12 at 2
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Motion at Constant Acceleration
Special case when a does not change with time
Notation:
tf = t ti = 0
xf = x xi = xo
vf = v vi = vo
a=
vavg =
v f − vi
t f − ti
x f − xi
a=
v − vo
t
“t at time zero”
“location at time zero”
“velocity at time zero”
v = vo + at
x − xo
x = xo + vavg t
t f − ti
t
Similar derivations lead to more equations:
vavg =
v + vo
2
vavg =
Δx = vot + 12 at 2 v 2 = vo 2 + 2aΔx
COMMON EQUATIONS OF MOTION. You will use these in homeworks and on test. They will be GIVEN on test!
[future: could maybe fit a mini-lecture on this before pre-test?]
Which formula to use?
vavg
v − vo
=
2
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Which formula to use?
v = vo + at
v 2 = vo 2 + 2aΔx
v − vo
=
2
Δx = vot + 12 at 2
vavg
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Which formula to use?
v = vo + at
v 2 = vo 2 + 2aΔx
v − vo
=
2
Δx = vot + 12 at 2
vavg
Pro Tip #3: List what you know and need to know in variable form
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Which formula to use?
v = vo + at
v 2 = vo 2 + 2aΔx
v − vo
=
2
Δx = vot + 12 at 2
vavg
Pro Tip #3: List what you know and need to know in variable form
• 1 equation with one unknown is solvable.
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Which formula to use?
v = vo + at
v 2 = vo 2 + 2aΔx
v − vo
=
2
Δx = vot + 12 at 2
vavg
Pro Tip #3: List what you know and need to know in variable form
• 1 equation with one unknown is solvable.
• 2 equations with two unknowns is solvable.
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Which formula to use?
v = vo + at
v 2 = vo 2 + 2aΔx
v − vo
=
2
Δx = vot + 12 at 2
vavg
Pro Tip #3: List what you know and need to know in variable form
• 1 equation with one unknown is solvable.
• 2 equations with two unknowns is solvable.
Pro Tip # 4: Practice helps you pick best formulas!
A lot of the work will be deciding what information to plug in to which equation.
If I know my final v, initial v, and final time, I can compute acceleration. If I don’t know my final acceleration or time, I can’t compute them based on the information given.
Let’s Practice!
The speed of a nerve impulse in the human body is about
100 m/s. If you accidentally stub your toe in the dark,
estimate the time it takes the nerve impulse to travel to
your brain.
This is also why it seems to take too long to get your hand off of a hot surface. You’re already a little burned before you can react to take your hand away!
By the way, never ever type “stubbed toe” into google images, it comes up with seriously the worst stuff.
Let’s Practice!
The speed of a nerve impulse in the human body is about
100 m/s. If you accidentally stub your toe in the dark,
estimate the time it takes the nerve impulse to travel to
your brain.
Draw a picture and list knowns and unknowns
This is also why it seems to take too long to get your hand off of a hot surface. You’re already a little burned before you can react to take your hand away!
By the way, never ever type “stubbed toe” into google images, it comes up with seriously the worst stuff.
Let’s Practice!
The speed of a nerve impulse in the human body is about
100 m/s. If you accidentally stub your toe in the dark,
estimate the time it takes the nerve impulse to travel to
your brain.
Draw a picture and list knowns and unknowns
Average velocity = 100 m/s = displacement / time
This is also why it seems to take too long to get your hand off of a hot surface. You’re already a little burned before you can react to take your hand away!
By the way, never ever type “stubbed toe” into google images, it comes up with seriously the worst stuff.
Let’s Practice!
The speed of a nerve impulse in the human body is about
100 m/s. If you accidentally stub your toe in the dark,
estimate the time it takes the nerve impulse to travel to
your brain.
Draw a picture and list knowns and unknowns
Average velocity = 100 m/s = displacement / time
Change in time = Δt = Δx/v = ~2 m / 100 m/s
This is also why it seems to take too long to get your hand off of a hot surface. You’re already a little burned before you can react to take your hand away!
By the way, never ever type “stubbed toe” into google images, it comes up with seriously the worst stuff.
Let’s Practice!
The speed of a nerve impulse in the human body is about
100 m/s. If you accidentally stub your toe in the dark,
estimate the time it takes the nerve impulse to travel to
your brain.
Draw a picture and list knowns and unknowns
Average velocity = 100 m/s = displacement / time
Change in time = Δt = Δx/v = ~2 m / 100 m/s
= 0.02 s or 20 milliseconds
This is also why it seems to take too long to get your hand off of a hot surface. You’re already a little burned before you can react to take your hand away!
By the way, never ever type “stubbed toe” into google images, it comes up with seriously the worst stuff.
Problems inside problems
Might need to break down problem into
smaller pieces! Solve in sequence.
Let’s Practice!
1 mile = 1609 m
Let’s Practice!
A rocket ship is capable of accelerating at a
rate of 0.60 m/s2. How long does it take for it
to get from going 55 mi/h to going 60 mi/h?
1 mile = 1609 m
Let’s Practice!
A rocket ship is capable of accelerating at a
rate of 0.60 m/s2. How long does it take for it
to get from going 55 mi/h to going 60 mi/h?
Draw a picture and list knowns and unknowns
1 mile = 1609 m
Let’s Practice!
A rocket ship is capable of accelerating at a
rate of 0.60 m/s2. How long does it take for it
to get from going 55 mi/h to going 60 mi/h?
Draw a picture and list knowns and unknowns
Want: Δt Know: vo, vf, a
1 mile = 1609 m
Let’s Practice!
A rocket ship is capable of accelerating at a
rate of 0.60 m/s2. How long does it take for it
to get from going 55 mi/h to going 60 mi/h?
Draw a picture and list knowns and unknowns
Want: Δt Know: vo, vf, a
v = vo +a Δt rearrange: Δt = (v-vo)/a
1 mile = 1609 m
Let’s Practice!
A rocket ship is capable of accelerating at a
rate of 0.60 m/s2. How long does it take for it
to get from going 55 mi/h to going 60 mi/h?
Draw a picture and list knowns and unknowns
Want: Δt Know: vo, vf, a
v = vo +a Δt rearrange: Δt = (v-vo)/a
Will need to convert mi/h to what?
1 mile = 1609 m
While chasing its prey in a short sprint, a
cheetah starts from rest and runs 45 m in a
straight line, reaching a final speed of 72 km/h.
(a) Determine the cheetah’s average
acceleration during the short sprint, and (b) find
its displacement at t = 3.5s.
Problem Solving Pro-tips
1. Draw a picture!
2. Use and label your reference frame.
3. List what you KNOW and DON’T KNOW in
variable form.
4. Practice helps you pick best formulas!
REMINDER!