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Lesson

- Vectors Review

Scalars vs Vectors

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

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.

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

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.

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

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.

- 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

(No Transcript)

Lesson

- Unit Vectors

Consider Three Dimensions

Polar Angle

z

Azimuthal Angle

az

q

ay

y

f

ax

xy Projection

x

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.

Unit Vectors

z

k

j

i

y

x

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

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

- 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?

Lesson

- 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

- 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

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

Lesson

- 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) 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.

- 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

- 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

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.

Position graphs for 2-D projectiles. Assume

projectile fired over level ground.

Velocity graphs for 2-D projectiles. Assume

projectile fired over level ground.

Vy

Vx

t

t

Acceleration graphs for 2-D projectiles. Assume

projectile fired over level ground.

ay

ax

t

t

RememberTo work projectile problems

- resolve the initial velocity into components.

Vo

?

- 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?

Lesson

- Monkey Gun Experiment shooting on an angle

Lesson

- A day of derivations

- Sample problem derive the trajectory equation.

- Sample problem Derive the range equation for a

projectile fired over level ground.

- Sample problem Show that maximum range is

obtained for a firing angle of 45o.

- 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.

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!

Lesson

- Punt-pass-kick lab

Lesson

- Review of Uniform Circular Motion
- Radial and Tangential Acceleration

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.

Vectors inUniform Circular Motion

a v2 / r

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?

- 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?

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.

Tangential Acceleration

If tangential acceleration exists, either the

speed or the radius must change. This is no

longer UCM.

- 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?

- 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 when its

speed is 22 m/s?

Lesson

- Relative Motion

Derivation

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

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.

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.

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

The velocity measured by two observers depends

upon the observers velocity relative to each

other.

vA

vB

vB vA vrel vA vB vrel

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

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!

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.

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

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.

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