Sept 9: Chapter 2
Describing Motion: Kinematics in One Dimension
Warning: former students say kinematics is hard at
first………..but by December they say it is easy
First, a question
• You walk at 0.80 m/s through the woods.
How long will it take for you to travel 2.444
km?
a. Report in seconds.
b. Report in hours.
Manage sig figs.
Managing homework
• If you don’t have 75% of your assignment
done…
– Give yourself a HW check
– Write date, name, and assignment you
missed in “the purple notebook”
• If, two classes later, you haven’t done it,
you’ll get one more HW check
Sections of Chapter 2 we will cover…
•Reference Frames and Displacement
•Average Velocity
•Instantaneous Velocity
•Acceleration
•Motion at Constant Acceleration
•Solving Problems
•Falling Objects
•Graphical Analysis of Linear Motion
Speed of the bowling ball
• Using only what you have, estimate the
speed as I roll it.
• What is the velocity?
•
It is 100,000 km/hr! Yes, it is.
• What is the acceleration?
• We need to define our reference for
measurement.
2-1 Reference Frames
Any measurement of position, distance, or
speed must be made with respect to a
reference frame.
For example, if you are sitting on a train and someone
walks down the aisle, their speed with respect to the
train is a few miles per hour, at most. Their speed with
respect to the ground is much higher.
2-1 Displacement
We make a distinction between distance and
displacement.
Displacement (blue line) is how far the object is
from its starting point, regardless of how it got
there.
Distance traveled (dashed line) is measured
along the actual path taken.
We will use both.
2-1 Displacement
The displacement is written:
Left:
Right:
Displacement is positive.
Displacement is negative.
2-2 Average Speed & Velocity
Speed: how far an object travels in a given time
interval
(2-1)
Velocity includes directional information:
Check: *can avg speed be lower than avg velocity?
* What is slowest avg speed? Highest? …
check
• Yesterday, this book was at my desk, and
today it is here. Calculate average v.
• North America drifts about 1 inch per year.
– What is average velocity in SI units?
2-3 Instantaneous Velocity
The instantaneous velocity is like the average
velocity, in the limit as the time interval becomes
infinitesimally short.
(2-3)
These graphs show (a)
constant velocity and (b)
varying velocity.
Check: graph my acceleration.
Velocity with go-motion
• Observe this cart motion and estimate the
instantaneous velocity in SI units.
• Now check it with go-motion device.
Let’s talk graphs and carts
• A car on a track, x is position from origin
• What is positive, what is negative?
• Now you Plot x vs. t, identify origin (0,0)
–
–
–
–
X is plotted on vertical but represents horizontal !?!?
What does slope tell us?
What is velocity, what does it look like?
Think about physical motion, and think about
graphical representation of that motion.
• Check: given constant velocity, what is a?
Make graphs from motion
1. bowling ball on ground
2. Fan car on ground
3. Fan car up a sloped table, no
motion
4. Bowling ball down sloped table
5. Bowling ball up a sloped table
b
a
x
x
t
t
d
c
v
x
t
t
Take the motion graph
challenge on loggerpro
• Loggerpro
FILE/OPEN/Physics/graphmatching
• 1b
• 1c
• 1d
• 1e velocity
• 1g velocity
*2-4 Acceleration
Acceleration is the rate of change of velocity.
(How fast is velocity changing)
2-4 Acceleration
Acceleration, like velocity is a vector, although in onedimensional motion we only need the sign.
The previous image shows positive acceleration; here
is negative acceleration with positive velocity:
2-4 more Acceleration
There is a difference between negative acceleration and
deceleration:
Negative acceleration is acceleration in the negative
direction as defined by the coordinate system.
Deceleration occurs only the acceleration is opposite in
direction to the velocity.
I recommend avoiding the term deceleration in this
class.
2-4 Acceleration
The instantaneous acceleration is the average
acceleration, in the limit as the time interval
becomes infinitesimally short.
(2-5)
It is a snapshot of the acceleration at a given
moment.
Check: graph my acceleration…
Acceleration check
• A car drives 25m/s and then accelerates to
45m/s in 10.0 seconds.
• What is average acceleration?
• In the morning, I drive my car from our
garage to the school and park. My
maximum speed is 45mph and the trip
takes me 10 minutes. What is average
acceleration of my car?
Special case: Motion at Constant
Acceleration (the math that works for us)
We can write the average velocity of an object
during a time interval t as
What do
these all
mean?
The acceleration, now assumed or known to be
constant, is
Check: estimate the a for this motion…
2-5 Motion at Constant Acceleration,
deriving the useful equations.
In addition, as the velocity is increasing at a
constant rate, we know that the average is:
(2-8)
Combining these last three equations, we find:
(2-9)
Note: I will not expect you to derive these
kinematic equations…
2-5 Motion at Constant Acceleration, the
kinematic equations
We can also combine these equations so as to
eliminate t:
(2-10)
We now have all the equations we need to solve
constant-acceleration problems.
(2-11a)
(2-11b)
(2-11c)
(2-11d)
A typical (classic) problem
• You push gently on the accelerator of your
brand new car, and the car moves from a
stop to 22m/s in 63 m. Find a.
• Draw at beginning and at end = 2 pictures
• What do you know at each time?
• What equation will use this info and give
you a?
• Solve symbolically for a and check units.
• Now calculate the number and sig figs.
• Does your answer make sense?
*“The Big 7”
Problem solving steps
1. Draw and label the situation at two times
1. Write something for xo, x, to, t, vo, v, and a
2. Decide and show which direction is +
2. Find the correct kinematic equation
3. Delete what is zero
4. Solve algebra for desired unknown (do
one unknown at a time)
5. Check the units of the algebraic equation
6. Plug in the numbers and units, calculate.
7. Fix answer for correct sig figs
2-6 Solving Problems (Big 7 are in red)
1. Read the whole problem and make sure you
understand it. Then read it again.
2. Decide on the objects under study and what
the time interval is.
3. Draw a diagram and choose coordinate axes.
4. Write down the known (given) quantities, and
then the unknown ones that you need to find.
5. What physics applies here? Plan an approach
to a solution.
2-6 Solving Problems (cont)
6. Which equations relate the known and
unknown quantities? Are they valid in this
situation? Solve algebraically for the unknown
quantities, and check that your result is sensible
(correct dimensions).
7. Calculate the solution and round it to the
appropriate number of significant figures.
8. Look at the result – is it reasonable? Does it
agree with a rough estimate?
9. Check the units again.
Constant acceleration
•
•
•
•
It is one number
It is one value
It can be zero
What are examples?
Demo with fan car and
GoMotion
• Measure x, v, a
• Calculate a and check it
2-7 Falling Objects
Near the surface of the Earth, all objects
experience approximately the same acceleration
due to gravity.
This is one of the most
common examples of
motion with constant
acceleration.
We love it.
2-7 Falling Objects
In the absence of air
resistance, all objects
fall with the same
acceleration, although
this may be hard to tell
by testing in an
environment where
there is air resistance.
Demo: book and paper,
vacuum and feather
• Sketch the set up and show me
• Write the procedure in 4-5 bullet points
• write these answers
– What happens to the air in the chamber?
– Why do paper and feather fall slowly?
– Why does magnet fall quickly?
– Why does feather fall slower than paper?
– Why do meteorites burn up in the
atmosphere?
– Can anything fall in air without any friction?
2-7 Falling Objects, the data
The acceleration due to
gravity at the Earth’s
surface is approximately
9.80 m/s2.
How to solve vertical motion
kinematics?
•Same as horizontal!
•And know that a = -g
when up is positive
Look at HW problems
• Practice together
• Do example 2-5
• And do p-27 together in class
15 sept: More on graphs
• Plots of x vs. t or v vs. t
– Look at 6 cases on next page
– Describe motion of each
– Derive companion graphs from each
– Always do graph down first
– Add numbers if it helps
– Be sure you can understand units of each
x
t
x
t
x
t
v
t
v
t
v
t
a
t
a
t
a
t
x
t
x
t
x
t
v
t
v
t
v
t
a
t
a
t
a
t
Quiz on motion
• A fan car has constant acceleration from
4.0m/s to 1.0 m/s in 2.0 m.
– What is a?
– What is tf?
– Plot the motion graphs x, v, and a without
numbers
Understanding x, v, and a
• http://phet.colorado.edu/en/simulation/moving-man
• To understand motion, we need to
understand the graphs of x, v, and a vs. t,
and know how those relate to actual
motion.
• Check
– X,v,a, of no motion, constant position
– Xva of constant velocity
– Xva of constant acceleration
*Check your
“graphs understanding”
• Draw x,v,a vs. t for each motion
• Motions:
x
– fan car push away
– Ball toss up
– Ball bounce
– Falling coffee filter
t
v
t
a
t
Now make it real
• Write each of these motion graphs on a
page you can carry.
• Go into the hall and practice each motion
as a table, reach consensus.
• I will quiz you in 5 minutes using
Loggerpro and this cart.
• Each group will match the graph
on Loggerpro
2-8 Graphical Analysis of Linear Motion
You’ve done this in math?
This is a graph of x vs. t
for an object moving with
constant velocity. The
velocity is the slope of the
x-t curve.
Graphical analysis made easy
6m/s
v
Area under the v-t curve =
the distance traveled
0
t
5s
• How far did this object go in 5s?
• What is the area under this curve?
• The same is true for any shape of v-t
curve!
2-8 Graphical Analysis of Linear Motion
On the left we have a graph of velocity vs. time
for an object with varying velocity; on the right
we have the resulting x vs. t curve. The
instantaneous velocity is tangent to the curve at
each point.
2-8 Graphical Analysis of Linear Motion
Don’t need this yet!
The displacement, x,
is the area beneath
the v vs. t curve.
Quiz on a classic problem
• I stand on the school roof and drop a rock
from 6.0 m.
– Draw a picture.
– Write in all known and unknown values for t,
y, v, and a.
– How long does it take to hit the ground?
– How fast is it going when it hits?
Summary of Chapter 2
• Kinematics is the description of how objects
move with respect to a defined reference frame.
• Displacement is the change in position of an
object.
• Average speed is the distance traveled divided
by the time it took; average velocity is the
displacement divided by the time.
• Instantaneous velocity is the limit as the time
becomes infinitesimally short.
Summary of Chapter 2 cont.
• Average acceleration is the change in velocity
divided by the time.
• Instantaneous acceleration is the limit as the
time interval becomes infinitesimally small.
• The equations of motion for constant
acceleration are given in the text; there are four,
each one of which requires a different set of
quantities.
• Objects falling (or having been projected) near
the surface of the Earth experience a gravitational
acceleration of 9.80 m/s2.
•We can solve problems using 7 steps…
Practice during single sessions: 16 P 26 and 57
Classics chapter 2
1. Simple horizontal acceleration; fan car
2. Ball off a cliff, with v0 = 0 (use newton ball
to demo)
3. Ball off a cliff, but with initial velocity down
4. Ball thrown up and falls down
5. Car braking to a stop, how far, or time
6. Car into tree, front compresses, what is
a?
7. Guy falls into net, decelerates, two parts
8. The 6 graphs of x, v, a vs t
Comments on exam 1
• Physics (and Chem) are often the first
classes where some students get grades
less than 90%. Was true for me.
• Don’t panic on this test.
• You have seen all the techniques: algebra,
vectors, solving problems
• You just need practice.
• You can do this.
Exam 1 post mortem
– Mean = average = 89%
– Low=60; high = 100
• Review with different pen
• Record correct answers
• Let me know after class if addition or
correction mistake by me
• Take exam and correct any answer for
Friday. Make this legible. Attach sheet to
show new work.
Lab 1: velocity and acceleration
of a bowling ball
• 3 per group.
• Share a ball
to get set
up.
• Find or build
a ramp.
5 min: agree on plan,
assign tasks
5 min: set up and take a
trial run with timers
15 min: conduct runs with
GoMotion and timers
10 min: short present to
Dr. F of results. Save data
5 min: clean up
TC marathon