Vectors Review

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Lesson

• Vectors Review

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Scalars vs Vectors

• Scalars have magnitude only
• Distance, speed, time, mass
• Vectors have both magnitude and direction
• displacement, velocity, acceleration

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Direction of Vectors

• The direction of a vector is represented by the
  direction in which the ray points.
• This is typically given by an angle.

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Magnitude of Vectors

• The magnitude of a vector is the size of whatever
  the vector represents.
• The magnitude is represented by the length of the
  vector.
• Symbolically, the magnitude is often represented
  as A

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Equal Vectors

• Equal vectors have the same length and direction,
  and represent the same quantity (such as force or
  velocity).

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Inverse Vectors

• Inverse vectors have the same length, but
  opposite direction.

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Graphical Addition of Vectors

• Vectors are added graphically together
  head-to-tail.
• The sum is called the resultant.
• The inverse of the sum is called the equilibrant

A B R
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Component Addition of Vectors

• Resolve each vector into its x- and y-components.
• Ax Acos? Ay Asin?
• Bx Bcos? By Bsin? etc.
• Add the x-components together to get Rx and the
  y-components to get Ry.
• Use the Pythagorean Theorem to get the magnitude
  of the resultant.
• Use the inverse tangent function to get the angle.

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• Sample problem Add together the following
  graphically and by component, giving the
  magnitude and direction of the resultant and the
  equilibrant.
• Vector A 300 m _at_ 60o
• Vector B 450 m _at_ 100o
• Vector C 120 m _at_ -120o

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Lesson

• Unit Vectors

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Consider Three Dimensions
Polar Angle
z
Azimuthal Angle
az
q
ay
y
f
ax
xy Projection
x
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Unit Vectors

• Unit vectors are quantities that specify
  direction only. They have a magnitude of exactly
  one, and typically point in the x, y, or z
  directions.

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Unit Vectors
z
k
j
i
y
x
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Unit Vectors

• Instead of using magnitudes and directions,
  vectors can be represented by their components
  combined with their unit vectors.
• Example displacement of 30 meters in the x
  direction added to a displacement of 60 meters in
  the y direction added to a displacement of 40
  meters in the z direction yields a displacement
  of

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Adding Vectors Using Unit Vectors

• Simply add all the i components together, all the
  j components together, and all the k components
  together.

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• Sample problem Consider two vectors, A 3.00 i
  7.50 j and B -5.20 i 2.40 j. Calculate C
  where C A B.

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• Sample problem You move 10 meters north and 6
  meters east. You then climb a 3 meter platform,
  and move 1 meter west on the platform. What is
  your displacement vector? (Assume East is in the
  x direction).

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Suppose I need to convert unit vectors to a
magnitude and direction?

• Given the vector

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• Sample problem You move 10 meters north and 6
  meters east. You then climb a 3 meter platform,
  and move 1 meter west on the platform. How far
  are you from your starting point?

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Lesson

• Position, Velocity, and Acceleration Vectors in
  Multiple Dimensions

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1 Dimension 2 or 3 Dimensions

• x position
• ?x displacement
• v velocity
• a acceleration

• r position
• ?r displacement
• v velocity
• a acceleration

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• Sample problem The position of a particle is
  given by r (80 2t)i 40j - 5t2k. Derive the
  velocity and acceleration vectors for this
  particle. What does motion look like?

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• Sample problem A position function has the form
  r x i y j with x t3 6 and y 5t - 3.
• a) Determine the velocity and acceleration
  functions.
• b) Determine the velocity and speed at 2 seconds.

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Miscellaneous

• Lets look at some video analysis.
• Lets look at a documentary.
• Homework questions?

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Lesson

• Multi-Dimensional Motion with Constant (or
  Uniform) Acceleration

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• Sample Problem A baseball outfielder throws a
  long ball. The components of the position are x
  (30 t) m and y (10 t 4.9t2) m
• a) Write vector expressions for the balls
  position, velocity, and acceleration as functions
  of time. Use unit vector notation!
• b) Write vector expressions for the balls
  position, velocity, and acceleration at 2.0
  seconds.

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• Sample problem A particle undergoing constant
  acceleration changes from a velocity of 4i 3j
  to a velocity of 5i j in 4.0 seconds. What is
  the acceleration of the particle during this time
  period? What is its displacement during this time
  period?

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Trajectory of Projectile

• This shows the parabolic trajectory of a
  projectile fired over level ground.
• Acceleration points down at 9.8 m/s2 for the
  entire trajectory.

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Trajectory of Projectile
vx
vx
vy
vy
vx
vy
vx
vx
vy

• The velocity can be resolved into components all
  along its path. Horizontal velocity remains
  constant vertical velocity is accelerated.

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Position graphs for 2-D projectiles. Assume
projectile fired over level ground.
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Velocity graphs for 2-D projectiles. Assume
projectile fired over level ground.
Vy
Vx
t
t
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Acceleration graphs for 2-D projectiles. Assume
projectile fired over level ground.
ay
ax
t
t
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RememberTo work projectile problems

• resolve the initial velocity into components.

Vo
?
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• Sample problem A soccer player kicks a ball at
  15 m/s at an angle of 35o above the horizontal
  over level ground. How far horizontally will the
  ball travel until it strikes the ground?

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• Sample problem A cannon is fired at a 15o angle
  above the horizontal from the top of a 120 m high
  cliff. How long will it take the cannonball to
  strike the plane below the cliff? How far from
  the base of the cliff will it strike?

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Lesson

• Monkey Gun Experiment shooting on an angle

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Lesson

• A day of derivations

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• Sample problem derive the trajectory equation.

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• Sample problem Derive the range equation for a
  projectile fired over level ground.

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• Sample problem Show that maximum range is
  obtained for a firing angle of 45o.

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Lesson

• Relative Motion

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Relative Motion

• When observers are moving at constant velocity
  relative to each other, we have a case of
  relative motion.
• The moving observers can agree about some things,
  but not about everything, regarding an object
  they are both observing.

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Consider two observers and a particle. Suppose
observer B is moving relative to observer A.
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Also suppose particle P is also moving relative
to observer A.
In this case, it looks to A like P is moving to
the right at twice the speed that B is moving in
the same direction.
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However, from the perspective of observer B
vA
vB
it looks like P is moving to the right at the
same speed that A is moving in the opposite
direction, and this speed is half of what A
reports for P.
vrel
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The velocity measured by two observers depends
upon the observers velocity relative to each
other.
vA
vB
vB vA vrel vA vB vrel
vrel
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Sample problem Now show that although velocity
of the observers is different, the acceleration
they measure for a third particle is the same
provided vrel is constant. Begin with vB vA -
vrel
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Galileos Law of Transformation of Velocities

• If observers are moving but not accelerating
  relative to each other, they agree on a third
  objects acceleration, but not its velocity!

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Inertial Reference Frames

• Frames of reference which may move relative to
  each other but in which observers find the same
  value for the acceleration of a third moving
  particle.
• Inertial reference frames are moving at constant
  velocity relative to each other. It is impossible
  to identify which one may be at rest.
• Newtons Laws hold only in inertial reference
  frames, and do not hold in reference frames which
  are accelerating.

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Sample problem How long does it take an
automobile traveling in the left lane at 60.0km/h
to pull alongside a car traveling in the right
lane at 40.0 km/h if the cars front bumpers are
initially 100 m apart?
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Sample problem A pilot of an airplane notes that
the compass indicates a heading due west. The
airplanes speed relative to the air is 150 km/h.
If there is a wind of 30.0 km/h toward the north,
find the velocity of the airplane relative to the
ground.