Overview: Kinematics is our description of motion of AN OBJECT (a thing with properties), described by AN OBSERVER (a mythical being that decides where to place axes). Kinematics is a foundation or framework we use to help us describe laws of physics (later). Kinematics describes motion! Where is the object (position along an axis)? How far has the object moved in some time(displacement)? How fast (speed) and Which way (direction)? Both together make "velocity" Is the velocity constant (uniform motion)? What is the rate of change of velocity (acceleration)? Special case (constant acceleration). Turning point Properties of motion of an object: We consider motion (OF AN OBJECT) in one dimension-- along a single axis. The direction has only two choices: Positive (+) and Negative (-) or "right" and "left" respectively. The numerical results may depend on how the observer lays out axes. Physics does not change due to a different observer! Ch. 2 Kinematics in one dimension Notes 2 Ch 2 Kinematics Page 1
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Overview: Kinematics is our description of motion of AN OBJECT (a thing with properties), described by AN OBSERVER (a mythical being that decides where to place axes).
Kinematics is a foundation or framework we use to help us describe laws of physics (later). Kinematics describes motion!
Where is the object (position along an axis)?
How far has the object moved in some time(displacement)?
How fast (speed) and Which way (direction)? Both together make "velocity"
Is the velocity constant (uniform motion)?
What is the rate of change of velocity (acceleration)?
Special case (constant acceleration).
Turning point
Properties of motion of an object:
We consider motion (OF AN OBJECT) in one dimension--along a single axis. The direction has only two choices: Positive (+) and Negative (-) or "right" and "left" respectively.
The numerical results may depend on how the observer lays out axes.
Physics does not change due to a different observer!
Ch. 2 Kinematics in one dimension
Notes 2 Ch 2 Kinematics Page 1
Definitions:velocity=Rate of change of position. Note the distinction between Average Velocity, Instantaneous Velocity, Initial Velocity, Final Velocity
Acceleration= Rate of change of velocity.Note acceleration may be "zero", "uniform (constant)", or not-constant.
We can rearrange either velocity or acceleration equations to yield integral expressions for change in velocity or for displacement (change in position).
Note that in general eq 0
For the case of constant acceleration we get the following kinematic equation results that you must memorize, understand, and be able to use rapidly.
eq 1
eq 2
eq 3
eq 4
Sections 2.1 to 2.4 Definitions, constant accel. Equations
Notes 2 Ch 2 Kinematics Page 2
Displacement is "Change in position"
In one dimensional motion, the displacement may either be positive or negative(+ or -). There are only two choices for direction.
Equation "0" above gives a GENERAL definition for average velocity. We may be more specific and define instantaneous velocity:
SI or metric untis are "m/s" (meters per second). How big is "dt"?Consider one of your first trips in a car:
How often might you want an update of the velocity?
How about "Buster" in your passenger seat (Mythbusters)?
VOLUNTEERS: ok--JUST DO WHAT I TELL YOU.
When is say 0.500s time interval sufficient, when is it not sufficient?
Displacement, Velocity,
Notes 2 Ch 2 Kinematics Page 3
Describe acceleration, velocity, displacement during each interval.
•
Is acceleration constant?•What about average velocity during entire trip?•
etc other time intervals○
Average velocity from 0 to t1?•
Displacements each part of trip, overall?•What situation could make this motion?•
What questions can we ask about this motion?
Take a derivative to get v(t)Take another derivative to get a(t)
What can we do with this plot/information
Velocity vs. Time
Notes 2 Ch 2 Kinematics Page 4
How Fast? (speed)
Which way? (direction)
Is a vector. Can be expressed in either cartesian
(vx,vy) or polar (v,)KNOW HOW TO GET BACK AND FORTH.
=Rate of change of position (vector)
Velocity answers two questions
Also a vector. Rate of change of velocity (vector).Acceleration:
x vs. t (INTEGRAL)a vs. t (DERIVATIVE OF v vs. t)
From v vs. t plot we can get:
=t-to
=t-0
There is: t=tf-ti
Different t's:
In words:
Notes 2 Ch 2 Kinematics Page 5
Constant Acceleration
Remember those equations:
eq 1
eq 2
eq 3
eq 4
Eq1 From Definition of acceleration (rearrange)
Eq2 Halfway up the hill--
Eq 3 Relates how far to how long(how much time) given initial conditions
Eq 4 Relates how fast to how far (given initial conditions)
We can derive these equations either from our definitions of velocity and acceleration, or we can examine graphical methods (more instructive). CONSTANT ACCELERATION
v
t
vo
On this graph the slope (derivative) has meaning of acceleration (and is constant).
What does the area under the curve mean?
For constant slope, the velocity at any time is simply given by a straight line.
v(t)=vo+at where t means t final, and vo is at ti=0.
An Important SPECIAL CASE
Notes 2 Ch 2 Kinematics Page 6
For the graph we had above, the area under the curve from
0.00 to t has meaning of displacement x.
Graphically
=vot + (1/2)a t2
Note that I have changed up notation slightly from previously written equations, but the meaning is the same.
x=area of box plus area of triangle. What? Huh? Area
v
t
vo
The dashed line is the average velocity for the given time interval from 0 to tf
Note it is halfway between vo and vf
ONLY IF ACCELERATION IS CONSTANT
Again be flexible with v(t),
v, vf or vi and vo
Constant acceleration continues
Notes 2 Ch 2 Kinematics Page 7
So How do we get v2 equation from graph?
We don't--not directly. The graph was v vs. t.
We put together results and equation zero.
Rearrange this as:
Now for t plug-in
And
Result is the v2 equation
I call one "one half a t squared"I call another "v squared"I call another "average velocity"I call another "definition of accel"
These "names" each have unique meaning.
Naming the equations:
For cases of constant acceleration we can use these equations. NOTE THE NAMES MAY CHANGE---WE MAY USE
y rather than x or we may have ay and ax in a single problem. WHAT MAKES ACCELERATION HAPPEN?
Notes 2 Ch 2 Kinematics Page 8
You are moving 30.0m/s (~67 mph) and pass a state trooper. The trooper starts 1.00s later from rest and accelerates at 3.00m/s2.
Note we have TWO objects (trooper, car).
Many, many possible questions from the situation:
How far apart after 10.0s (from when)?Where does the trooper catch the car?When does the trooper catch the car?How fast is the trooper going when passing car?When is trooper moving your speed?
THINK ABOUT ALL THE QUESTIONS. etc.
Let's not pick a question yet. First decide what information we are given.
Where and how fast is everyone moving at t=0? And when is "t=0" anyway?
I am picking to start my stopwatch when the trooper starts moving (WHY---WHY---WHY)?
xc,o=30.0m vc,o=30.0m/s ac=0.00
xtr,o=0.00m vtr,o=0.00 atr=3.00m/s2
OK. So we have the initial information about both objects. Now let's pick a question or two.
When does the trooper pass the car (what time)?And Where does this happen? (location)
Trooper and the car:
Notes 2 Ch 2 Kinematics Page 9
xtr(tcatch)=xcar(tcatch) find tcatch (a particular time) that puts the trooper and car at the same place. Position
time
Setup the equation (I won't solve for you, but show you how to set up).
Solve for time. This is a quadratic so there are two solutions. Which works? What do they mean?
Trooper and car II
Notes 2 Ch 2 Kinematics Page 10
If acceleration is not constant then we have more general statements than our four kinematic equations and must go back to things like.
Dimensions, unitsI tend to stick to metric:
Length in meters (m)
How fast in meters/sec (m/s)
Acceleration in (m/s2)
I do make use of metric prefixes, but rarely use things like miles, LY,
and
Some calculus side notes
Notes 2 Ch 2 Kinematics Page 11
We typically replace x with y to describe vertical motion. So we are talking about throwing a ball straight up, or releasing a ball and letting it fall.
vo
y changesvy changesDoes ay change? What is ay?Does the ball reach a max height (called……)?
The ball moves up: Here are some things that happen:
We use the same old kinematic equations to describe the motion of the ball in such cases.
Is "ay" really uniform (constant everywhere)? When, when not?
Free fall Vertical one dim. motion
Notes 2 Ch 2 Kinematics Page 12
So, it appears that "g" does change if I get far away from the Earth compared to something…..WHAT?
Do we? Do we get far away very often?
OK. Let's stay grounded, right here on Earth.
y=0.00
y=50.0mTime to groundSpeed at groundPeak heightAccel at peakAverage v on way to peak
Some questions:
vy,o=+20.0m/s
The minus indicates direction. ay=-g (note g=9.80m/s2)
Given: yo=+50.0
Time to the peak (tpeak=?)Height of the peak (ypeak=?)
Displacement to peak (ypeak=?)---many other questions.
Find--answer:
Freefall II
Notes 2 Ch 2 Kinematics Page 13
Freefall III
Notes 2 Ch 2 Kinematics Page 14
Symmetry: If a ball goes up for some time on the way to the peak, and spends the same time coming down--it is moving the same speed, and has travelled the same distance.
The second solution to quadratics usually means something. The meaning may not be part of "this problem", but it could be part of another problem---so think about meaning.
