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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• Will the projectile always hit the target
  presuming it has enough range? The target will
  begin to fall as soon as the projectile leaves
  the gun.

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Punt-Pass-Kick Pre-lab

• Purpose Using only a stopwatch, a football
  field, and a meter stick, determine the launch
  velocity of sports projectiles that you punt,
  pass, or kick.
• Theory Use horizontal (unaccelerated) motion to
  determine Vx, and vertical (accelerated) motion
  to determine Vy. Ignore air resistance.
• Data Prepare your lab book to collect xi, xf,
  yo, and Dt measurements for each sports
  projectile. Analyze the data fully for at least
  three trials.
• Make sure you dress comfortably tomorrow!

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Lesson

• Punt-pass-kick lab

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Lesson

• Review of Uniform Circular Motion
• Radial and Tangential Acceleration

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Uniform Circular Motion

• Occurs when an object moves in a circle without
  changing speed.
• Despite the constant speed, the objects velocity
  vector is continually changing therefore, the
  object must be accelerating.
• The acceleration vector is pointed toward the
  center of the circle in which the object is
  moving, and is referred to as centripetal
  acceleration.

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Vectors inUniform Circular Motion
a v2 / r
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Sample Problem

• The Moon revolves around the Earth every 27.3
  days. The radius of the orbit is 382,000,000 m.
  What is the magnitude and direction of the
  acceleration of the Moon relative to Earth?

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• Sample problem Space Shuttle astronauts
  typically experience accelerations of 1.4 g
  during takeoff. What is the rotation rate, in
  rps, required to give an astronaut a centripetal
  acceleration equal to this in a simulator moving
  in a 10.0 m circle?

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Tangential acceleration

• Sometimes the speed of an object in circular
  motion is not constant (in other words, its not
  uniform circular motion).
• An acceleration component may be tangent to the
  path, aligned with the velocity. This is called
  tangential acceleration. It causes speeding up or
  slowing down.
• The centripetal acceleration component causes the
  object to continue to turn as the tangential
  component causes the speed to change. The
  centripetal component is sometimes called the
  radial acceleration, since it lies along the
  radius.

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Tangential Acceleration
If tangential acceleration exists, either the
speed or the radius must change. This is no
longer UCM.
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• Sample Problem Given the figure at right
  rotating at constant radius, find the radial and
  tangential acceleration components if q 30o and
  a has a magnitude of 15.0 m/s2. What is the speed
  of the particle at the location shown? How is the
  particles speed changing?

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• Sample problem Suppose you attach a ball to a 60
  cm long string and swing it in a vertical circle.
  The speed of the ball is 4.30 m/s at the highest
  point and 6.50 m/s at the lowest point. Find the
  acceleration of the ball at the highest and
  lowest points.

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• Sample problem A car is rounding a curve on the
  interstate, slowing from 30 m/s to 22 m/s in 7.0
  seconds. The radius of the curve is 30 meters.
  What is the acceleration of the car when its
  speed is 22 m/s?

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Lesson

• Relative Motion

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Derivation

• Why is a v2/r?
• Follow along, and see a classic derivation

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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.