Lesson

Vectors Review

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

Lesson

Unit Vectors

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?

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

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?

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) 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?

Lesson

Monkey Gun Experiment – shooting

on an angle

Lesson

A day of derivations

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

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 t 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 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?

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 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.

v

Tangential Acceleration

radial or centripetal component (ar or ac )

tangential component (aT )

aIf 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 = 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 particle’s speed changing?

5.00 ma

v

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.

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.