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Transcript of Wednesday, August 27, 2008 Vectors Review Announcements Lab fee? Chris, Liam, Greg? Lab books due...
Wednesday, August 27, 2008
Vectors Review
Announcements Lab fee? Chris, Liam, Greg? Lab books due today with
Kinematics Graphing lab.
Scalars vs Vectors Scalars have magnitude only
Distance, speed, time, mass Vectors have both magnitude and
direction displacement, velocity, acceleration
R
headtail
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.
Ax
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 │
AIf vector A represents a displacement of three miles to the north…
B
Then vector B, which is twice as long, would represent a displacement of six miles to the north!
Equal Vectors
Equal vectors have the same length and direction, and represent the same quantity (such as force or velocity).
Inverse Vectors Inverse vectors have the same length,
but opposite direction.
A
-A
A
B
RA + B = R
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
Component Addition of Vectors
1) Resolve each vector into its x- and y-components.Ax = Acos Ay = AsinBx = Bcos By = Bsin etc.
2) Add the x-components together to get Rx and the y-components to get Ry.
3) Use the Pythagorean Theorem to get the magnitude of the resultant.
4) Use the inverse tangent function to get the angle.
• 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 @ 60o
– Vector B: 450 m @ 100o
– Vector C: 120 m @ -120o
Thursday, August 28, 2008
Unit Vectors
Announcements Lab fee? Chris, Liam, Greg? HW Quiz: Chapter 2
Problems 30, 36, 39, 42, 49, 52 Roll the die!
Consider Three Dimensions
z
y
x
a
ax
ay
az
xy Projection
Polar Angle
Azimuthal Angle
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.
ˆ points in the x direction
ˆ points in the y direction
ˆ points in the z direction
i
j
k
Unit Vectors
z
y
x
ijk
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:
ˆˆ ˆ(30 -60 40 ) m
30,-60,40 m
i j k
Adding Vectors Using Unit Vectors
Simply add all the i components together, all the j components together, and all the k components together.
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.
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).
Suppose I need to convert unit vectors to a magnitude and direction?
Given the vector
2 2 2
ˆˆ ˆx y z
x y z
r r i r j r k
r r r r
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?
Friday, August 29, 2008
Position, Velocity, and Acceleration Vectors in
Multiple Dimensions
1 Dimension 2 or 3 Dimensions
x: position x: displacement v: velocity a: acceleration
r: position r: displacement v: velocity a: acceleration
r = x i + y j + z k r = x i + y j + z k v = vx i + vy j + vz k a = ax i + ay j + az k
In Unit VectorNotation
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”?
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.
Miscellaneous Let’s look at some video analysis. Let’s look at a documentary. Homework questions?
Tuesday, September 2, 2008
Multi-Dimensional Motion with Constant (or Uniform)
Acceleration
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) ma) Write vector expressions for the ball’s position, velocity, and acceleration as functions of time. Use unit vector notation!
b) Write vector expressions for the ball’s position, velocity, and acceleration at 2.0 seconds.
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?
Trajectory of Projectile
g
g
g
g
g
This shows the parabolic trajectory of a projectile fired over level ground.
Acceleration points down at 9.8 m/s2 for the entire trajectory.
Trajectory of Projectile
vx
vy
vy
vx
vx
vy
vx
vy
vx
The velocity can be resolved into components all along its path. Horizontal velocity remains constant; vertical velocity is accelerated.
Position graphs for 2-D projectiles. Assume projectile fired over level ground.
x
y
t
y
t
x
t
Vy
t
Vx
Velocity graphs for 2-D projectiles. Assume projectile fired over level ground.
Acceleration graphs for 2-D projectiles. Assume projectile fired over level ground.
t
ay
t
ax
Remember…To work projectile problems…
…resolve the initial velocity into components.
VoVo,y = Vo sin
Vo,x = Vo cos
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?
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?
Sample problem: derive the trajectory equation.
22 2
(tan )2 coso
gy x x
v
Sample problem: Derive the range equation for a projectile fired over level ground.
22 sin cosovRg
Sample problem: Show that maximum range is obtained for a firing angle of 45o.
22 sin cosovRg
Wednesday, September 3, 2008
Monkey Gun
Announcements Homework policy change.
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.
Friday, September 5, 2008
Review of Uniform Circular Motion
Uniform Circular Motion Occurs when an object moves in a circle
without changing speed. Despite the constant speed, the object’s
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.
Vectors inUniform Circular Motion
a
v
a = v2 / r
va
v
av
a
Sample ProblemThe 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?
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.
Wednesday, September 10, 2008
Radial and Tangential Acceleration
Tangential acceleration Sometimes the speed of an object in circular
motion is not constant (in other words, it’s not uniform circular motion).
An acceleration component is tangent to the path, aligned with the velocity. This is called tangential acceleration.
The centripetal acceleration component causes the object to continue to turn as the tangential component causes the radius or speed to change.
v
Tangential Acceleration
radial or centripetal component (ar or ac )
tangential component (aT )
aIf tangential acceleration exists, the orbit is not stable.
Sample Problem: Given the figure at right rotating at constant radius, find the radial and tangential acceleration components if = 30o and a has a magnitude of 15.0 m/s2. What is the speed of the particle? How is it behaving?
5.00 ma
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.
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?
Thursday, September 11, 2008
Relative Motion
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.
Consider two observers and a particle. Suppose observer B is moving relative to observer A.
Pparticle
Aobserver
Bobserver
vrel
Also suppose particle P is also moving relative to observer A.
Pparticle
Aobserver
Bobserver
vrel
vA
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.
However, from the perspective of observer B…
Pparticle
Aobserver
Bobserver
-vrel
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.
vA
vrel
The velocity measured by two observers depends upon the observers’ velocity relative to each other.
Pparticle
Aobserver
Bobserver
-vrel
vB
vB = vA – vrel
vA = vB + vrel
vA
vrel
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
Galileo’s Law of Transformation of Velocities If observers are moving but not
accelerating relative to each other, they agree on a third object’s acceleration, but not its velocity!
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.
Newton’s Laws hold only in inertial reference frames, and do not hold in reference frames which are accelerating.
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?
Sample problem: A pilot of an airplane notes that the compass indicates a heading due west. The airplane’s 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.