Speed Velocity and Acceleration in 1D and 2D

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Section 1.1-1.2

• Speed Velocity and Acceleration in 1D and 2D

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More Trig

• Pythagorean Theorem
• To find an angle, you need the inverse trig
  function
• i.e.

x y 1 What is r? ??
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Dynamics Kinematics

• Dynamics is the study of motion and of physical
  concepts
• (i.e. relationship between force and mass)
• Kinematics is a part of dynamics
• description of motion
• Not concerned with the cause of the motion

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Quantities in Motion

• Any motion involves three concepts
• Displacement
• Velocity
• Acceleration
• These concepts can be used to study objects in
  motion

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Section 1.1Displacement

• Defined as the change in position
• f stands for final
• i stands for initial
• SI units are meters (m)

Displacement vs. Time graph
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Displacement vs. Distance

• Displacement is NOT the same as Distance
• i.e. Throw a ball straight up and then catch it
  at the same point you released it
• The distance is twice the height
• The displacement is zero

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Vector Scalar Quantities

• Vector
• has magnitude direction
• i.e.
• - instantaneous velocity
• - instantaneous acceleration
• Scalar
• has magnitude only (v, t, mass)

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Speed

• Speed
• the total distance traveled divided by the total
  time elapsed
• Speed is a scalar quantity
• Average speed completely ignores any variations
  in the objects actual motion

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Average Velocity

• Velocity is the rate of change of displacement
  per unit time
• Units?

Slope is velocity!!!
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Speed vs. Velocity

• Cars on both paths have the same average
  velocity Why?
• they have the same displacement (in the same
  time interval).
• The car on the blue path will have a greater
  average speedWhy?
• the distance it travels is larger (in the same
  time interval)!

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Position vs. Time Graphs

• What will the position vs. time graph look like
  for the following
• Stand still
• Slow steady walk
• Fast steady walk
• How should I move to reproduce the graph on the
  previous slide?

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Interactive Position Time Graph

1. Stand still
2. Slow steady walk
3. Fast steady walk

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Acceleration

• the rate of change in velocity per unit time
• (?v/ ?t)
• Units?

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Graphical Interpretation of Average Acceleration

• Average acceleration equals the slope of the line
  joining the initial and final velocities (vs.
  time)

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Graphical Interpretation of Velocity

• Average velocity equals the slope of the line
  joining the initial and final positions (vs.
  time)

A start at 30m from start and decelerate (neg.
acceleration) to B
B Velocity is 0 m/s Start to accelerate in
reverse
C Constant velocity in reverse.
D At start position. Continue in reverse at
constant velocity
E Begin to accelerate in forward direction
(slow down in reverse). Stop at F.
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Quick Quiz

• Match each velocity vs. time graph to its
  corresponding acceleration vs. time graph.

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Motion Diagrams (Relationship between a and v)

• Uniform velocity
• What is the acceleration?
• a 0

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Relationship Between a and v

• v and a are in the same direction
• a is constant
• v is increasing

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Relationship Between a and v

• v and a are in opposite directions
• a is constant
• v is decreasing

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1. Which car or cars (red, green, and/or blue)
are undergoing an acceleration? Study each car
individually in order to determine the answer.
Red car moves at a constant velocity Green Car
accelerates Blue Car - accelerates
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2. Which car (red, green, or blue) experiences
the greatest acceleration?
Blue Car Has the greatest acceleration Green
Car Has the 2nd greatest acceleration Red car
does not accelerate
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3. Match the appropriate line to the particular
color of car.
Blue Car
Red car
Green Car
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Acceleration in Motion
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Animated Car Graphsd vs. t v vs. t a vs. t
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Drawing a dt vt and at graph

• Consider the data for a car experiencing uniform
  acceleration...

time (s) position (m)
0 0
0.5 0.25
1 1
1.5 2.25
2 4
2.5 6.25
3 9
3.5 12.25
4 16
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Drawing a dt vt and at graph

• We can calculate the instantaneous velocity at a
  point by drawing a tangent at that point.

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Drawing a dt vt and at graph

• By plotting these instantaneous velocities we can
  create a velocity time graph

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Drawing a dt vt and at graph

• From this vt graph we can create an acceleration
  time graph.

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Drawing a dt vt and at graph

• Displacement Time Graphs
• The Slope of a dt graph is the velocity
• Velocity Time Graphs
• The slope of a vt graph is acceleration
• The area under a vt graph is displacement
• Acceleration Time Graphs
• The area under a at graph is velocity

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Equality of Two Vectors

• Two vectors are equal if they have the same
  magnitude direction
• Are the vectors here equal?

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Vector Addition

• Given two vectors , what is
  ?

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Graphical Techniques of Vector Addition

• Tip-to-Tail Method
• Two vectors can be added by
• placing the tail of the 2nd on
• the tip of the 1st

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Multiplying a Vector by a Scalar

• Given , what is ?

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Graphical Techniques of Vector Addition

• What about subtraction?

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Quiz Question 1

• The vector c in the diagram is equal to
• 1.
• 2.
• 3.
• 4.
• 5. None of these

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Quiz Question 2

• The magnitudes of two vectors are 12
  units and 8 units, respectively. What are the
  largest and smallest possible values for the
  magnitude of the resultant vector
  ?
• 14.4 and 4
• 12 and 8
• 20 and 4
• None of these

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Components of a Vector
where and are the components of
the vector
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Components of a Vector
Notice also that
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Quiz Question 3

• The angle between where
  and the positive x axis
  is
• 29
• 61
• 151
• 209
• 241

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Quiz Question 4The Lawn Mower Question

• A lawn mower travelling 1.8 m/s E27S takes 4.5
  s to round a corner. It ends up travelling 1.8
  m/s N15E