INTRO TO KINEMATICS
The study of motion, regardless of cause
Average Speed
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As far back as 300-400 BC, the ancient Greeks
could calculate:
Speed = distance traveled
time elapsed
This is a scalar with SI units of m/s.
The difference between distance and
displacement
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Symbol: We will use x for displacement
Use d for distance, however be sure to distinguish it
from the d’s we use in calculus notation for differentials.
Distance is the length of the route traveled.
Displacement is how far the object traveled, i.e. “as the
crow flies”, in other words, it’s a vector.
Is the odometer on your car a distance or displacement
meter?
Is a quarterback concerned with the distance or
displacement of the receiver when he throws a pass?
Constant speed – a special situation
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Acceleration = 0
Some objects traveling with constant speed might
include a:
 Yo-yo
 Toy
car on a track
 Car on the highway with cruise control on
•
What does constant speed look like on a graph?
Exercise
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Sketch a speed of 10 m/s on a x-vs-t graph
What is the slope (derivative) of the x-vs-t graph?
What if the graph is not constant?
B
x (m)
D
P
C
A
t (s)
Instantaneous speed
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Speed is the slope of the d-vs-t graph at a given
point (a tangent line).
Which slope best represents the speed of the object
at point P?
Instantaneous speed = the limit as ∆t  0
or ∆d or even better, dx
∆t
dt
Calculus Break
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The rate of change at an instantaneous point on a
graph is called the derivative (it’s a tangent line).
Examples:
dx (2t + 1) = 2
dt
dx (41) = 0
dt
dx (t2) = 2t
dt
dx (sin t ) = cos t
dt
These are easily seen on a graph.
The derivative of the plot of x-vs-t is…
Velocity is expressed as a vector
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Recall:
 Add
vectors tip to tail
 Resultant vector is from start to finish
 You may add vectors in any order (Commutative
Property)
 You can move vectors as long as you don’t
 Rotate
them
 Change the length (magnitude)
More on velocity
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Velocity, as a vector, has speed and direction
Same units as speed (m/s), just not a scalar
Average velocity:
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v=x
t
Instantaneous velocity:
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v = dx
dt
Pitfalls in this unit to watch out for!
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Common intro error is to treat displacement,
velocity, acceleration equal. They are NOT the same
thing!
Use units to check.
Sign errors – signs indicate direction in vectors
Displacement ≠ distance
Calculus Break
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Recall that integral calculus is the “undo-ing” of
differential calculus.
So if the derivative of the x-vs-t graph is the speed
of the object, then the integral (area under the
curve) of the v-vs-t graph is the ____________.
Fundamental physics graphs we will use
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x-vs-t
v-vs-t
a-vs-t
Each one above is the previous plot, differentiated
Each one above is the plot that follows, integrated
This will make much more sense as we practice with
them.
TIPERs – Work in small groups
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NT3A-CT7
NT3A-RT9
NT3A-WWT10
NT3A-QRT20
NT3A-QRT21
NT3A-WWT22
Exit Ticket
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TIPER NT3A-WBT23
Rip out of your packet and put your name on it. Put in
the INBOX when you are finished. Show all work.
Should be done independently