Vector: a measurement that includes both magnitude and direction

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Vector a measurement that includes both
magnitude and direction
Scalar a measurement that has only a magnitude
(amount)
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Why are these important?What about objects that
more in multiple directions?
Consider a parked car, parked nose in, in a
parking spot. What gear do you put the car in
first if you want to leave the parking lot?
What direction does your car move? Is this
positive or negative? Then you put the car in
drive, or a low gear. What direction will you
move now? Positive or Negative?
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Practice Problem Consider a female student
seated at the center of the room. If she stands
up and walks to the front of the room 10 m away
and turns in an assignment, what is the total
distance traveled? What is her displacement?
Now consider the same female student. She
walks 10 m to the front of the room, turns in
her work and turns and walks back to the seat.
Now what is the total distance traveled? Now
tell me the distance the has traveled from her
seat. This is her displacement.
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So sometimes a vector measurement like
displacement will be the same as its scalar
counterpart, but sometimes it will be very
different.
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In the discovery assignment we learned the
following relationships between the mechanics
variables Time proceeds independently of
motion. Velocity is the change in displacement
divided by the time period during which that
change occurs. This is also called the rate of
change of displacement. Acceleration is the
change in velocity divided by the time period
during which that change occurs. This is also
called the rate of change of velocity. A rate
of change is the same thing as the slope on a
graph. So the SLOPE of the DISPLACEMENT graph
is the VELOCITY and the SLOPE of the VELOCITY
graph is the ACCELERATION.
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Sir Issac Newton developed an entire system of
mathematics to describe his observations of the
physical world. We call this mathematics CALCULUS
. Calculus allows us to calculate the
instantaneous rates of change of velocity and
acceleration.
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Average velocity Velocity over a period of time
Average acceleration Acceleration over a period
of time
As ?t approaches zero, the average rate of change
approaches the instantaneous rate of change.
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Motion GraphingWhy do we use/analyze graphs
representing motion?

• Graphs represent a relationship between two or
  more variable quantities.
• In mathematics you have treated variables as
  merely unknowns to be identified. In physics any
  variable is actually a quantity or a measurement
  of something real!

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1. In physics, as in science generally, one of the
   most studied types of relationships between
   variables is that of time dependency, which is
   easily demonstrated by a 2-D graph.
2. Often, the information that is highly evident on
   a graph of motion can only be proven
   quantitatively after many, and sometimes
   intensive calculations.

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GraphsAll graphs should have

• A title
• Clearly defined axes, labeled with measurement
  units.
• Gradations on the axes are usually helpful, but
  not necessary as long as any critical points are
  identifiable.
• NO LINES CONNECTING DATA POINTS unless the data
  is continuous.
• Thought given to what the slope (rise/run or
  y-values/x-values) means.

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• Independent variable(s) _________________________
  __________________________________________________
  _____________________________________________
• Dependent variable(s) ___________________________
  __________________________________________________
  ___________________________________________
• Which axis goes with each?
• When considering motion what do you think will be
  the most common independent variable?

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Motion Graphs

• Before we look at some real types of examples
  mentally recall the fundamental mechanics
  variables (x,v,a,t) and how they relate to each
  other. If youre not sure yet about that last
  part, hopefully it will become clear soon.
• One caveat Just because you are looking at a
  graph that may show, say, displacement vs. time,
  this does not preclude the fact that other
  information (about say, velocity or acceleration)
  may be available in that graph!

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Assume the origin represents a reference time
(t0) at a reference point in space
(x0). Identify and describe any critical points
on the graph. When describing the motion
represented on a graph during the time between
any consecutive critical points there are 2
questions to ask

1. Is the object moving forwards or backwards (with
   respect to the origin or starting point)?
2. Is the rate of movement the same? OR, said
   another way Is the object speeding up or slowing
   down?

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ANOTHER EXAMPLE Notice the slope of the
displacement graph its a constant because the
graph is a straight line, right?
x
v
a
t
t
t
Now look at the acceleration graphrecall that
acceleration is the slope, or rate of change, of
the velocity graph. What is the slope of a
horizontal line?
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Based on the displacement or velocity graphs
alone, it is possible to determine the other
characteristics of motion for the object.
x
t
Describe the behavior of the particle during
each time interval
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V
t
Describe the behavior of the particle during
each time interval
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Straight lines on a graph are easy. When curves
are present, it becomes a little harder to
interpret the motion.
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This graph looks familiar, but actually
represents motion that is very different than
the earlier graph of x vs. t.
v
t
t
On a velocity vs. time graph, the area under the
curve, or the integral, tells the displacement
of the object for the designated time interval.
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On an acceleration vs. time graph the area under
the graph tells the average velocity during that
time period. Notice how the area relationships
are sort-of opposite to the slope relationships
we discussed earlier. In calculus, the slope
(derivative) is the opposite of the area
(integral).
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Velocity
Acceleration
Description
- -
- -
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x
Slope
Area under graph
v
Slope
Area under graph
a
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Definitions commonly used in this class

• Displacement x
• Velocity v The average change in displacement
  over a given time interval. Commonly written as
  an equation
• Acceleration a The average change in velocity
  over a given time interval. Commonly written as
  an equation

Note the similarity to a slope calculation
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Basic Mechanics Problems

• Find the velocity of a car that accelerates at 10
  m/s2 from rest for 5 seconds.
• 2. Find the velocity of a thrown baseball that
  travels 75 m in a time of 1.5 s.
• 3. A girl starts at her front door, walks to her
  mailbox 30 m forward in a time of 5 seconds,
  pauses for 3 seconds, turns and walks back to the
  door in 4 seconds.
• a) What is her velocity during the trip to the
  mailbox?
• b) From the mailbox?
• c) During the whole trip?

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Free fall
Free fall is just a special case of motion in
which an object undergoes a constant
acceleration, in this case due to
gravity. Gavitational Acceleration has a
constant value that you must MEMORIZE.
Close to 10use 10 for approximating, but use 9.8
when you have a calculator.
The negative sign indicates direction.
Sometimes it is O.K. to ignore the sign, but you
must always be aware of and indicate the
direction of gravitational pull as downwards.
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Advanced mechanics equations
Recall
Assuming motion in only 1 direction, say forward,
from a relative zero position we can re-write
this equation as
Or more generally
Where x is the total displacement and x0 is the
initial (non-zero) displacement.
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Displacement under constant acceleration
If we add an acceleration factor, after a little
bit of calculus we get
If x0 is zero, this simplifies to
When dealing with free fall and later,
projectiles, displacement is in the vertical
direction. This equation allows us to easily
find the hang time of an object in the air.
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More on velocity
Recall
Assuming motion in only 1 direction, say forward,
from a relative zero initial velocity we can
re-write this equation as
Or more generally
Where v is the final velocity and v0 is the
initial (non-zero) velocity. t represents the
time interval in question.
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Last one
What if you dont know anything about time? Can
you still relate the variable? Surea little
algebraic manipulation of our other equations
and
Remember that x here is the displacement or net
distance. If you dont start from a zero
position, caution must be taken when using this
equation.
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So, given any combination of variables, x,v,a,
t, we can basically solve for any of the others,
provided that we know enough to begin with.
assumes a 0
doesnt need vfinal
doesnt need x
doesnt need t
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REMEMBERAcceleration is assumed to be
CONSTANTin order for these relationships to hold
true.
An acceleration of zero is a special case of a
constant acceleration.
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Practice

• A batter hits a baseball straight upwards with an
  initial velocity of 35 m/s. Find the maximum
  height of the ball, the time it takes to reach
  this height and the total time of flight.
• A car accelerates from rest at 5 m/s2. What is
  the velocity of the car as it passes the 55 m
  mark?
• What if the car had started rolling with an
  initial velocity of 3 m/s? Now what is the final
  velocity?
• A clay pigeon thrower tosses a clay bird 45 m
  straight upwards. If it took the pigeon 3 s to
  reach this height, find the initial velocity of
  the disk. (hint WATCH YOUR SIGNS)
• A madman throws a bowling ball upwards from the
  roof of a 40 m tall office building. If it
  impacts the ground with a velocity of 31 m/s,
  determine how high above the building it rose.

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Inertial Frames of Reference

• We often describe motion in terms of a static
  frame of reference. That is to say, we consider
  it as if we were a passive observer, standing
  stationary with respect to the action.
• Most real life situations involve dynamic, moving
  frames of reference. For instance, if you are
  driving a race car, you are probably not
  concerned with the speed of a passing car, but
  more likely concerned with how much faster it is
  traveling than you are. Using your inertial
  frame of reference this can be easily determined.

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• The frame of reference often serves as a starting
  point, used to determine initial, relative
  velocities. Once these are known, your
  kinematics equations kick in and everything
  proceeds normally.
• Caution Frames of reference are only important
  to consider when two moving objects are present,
  OR when a problem asks you to place yourself
  within the system defined by the problem (i.e.,
  use your imagination).

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Example

• You are a bug, clinging for life to a baseball
  that has just been wacked straight up. The ball
  reaches its maximum height and then
• Why are you on the ball? IRRELEVANT
• Why is the ball going straight up? B/c Most you
  cant handle a 2-D motion problem yetpatience.
• As the ball falls, describe the rate at which the
  ground approaches you.
• What is the rate at which you approach the ground?

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Example 2

• A miscreant teenager in the passenger side of a
  pickup truck (yee-haw!) fires a paintball gun at
  an old lady standing by her mailbox. If the
  truck is traveling at 25 m/s, and the paintball
  is fired at 90 m/s, neglecting air resistance
  what is the velocity of the paintball upon
  impacting the old woman?
• What is the velocity of the paintball with
  respect to the truck?

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• Same problem, paintball fired at an oncoming
  jogger, vjogger3 m/s. Find velocity of the
  paintball w/r to the jogger.
• Jogger going away at 3 m/s. Find velocity of
  paintball w/r to jogger.

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Trains, Planes and Automobiles

• Common problem type
• Two trains, A B, approach head on. Each train
  has velocity V as determined by a stationary
  observer. What is the velocity of train A
• With respect to the ground?
• With respect to train B?
• Two trains, A B, travel side-by-side. Train A
  has velocity V and train B has velocity 3V,
  determined by a stationary observer. What is the
  velocity of train A with respect to train B?