Chapter 3 Vectors and Two Dimensional Motion. Vectors Motion in 1D Negative/Positive for Direction ○ Displacement ○ Velocity Motion in 2D and 3D 2.

Chapter 3Vectors and Two Dimensional Motion

Vectors Motion in 1D

Negative/Positive for Direction○ Displacement○ Velocity

Motion in 2D and 3D2 or 3 displacements

○ Too much work○ Easier way to describe these motions

Vectors!Magnitude and direction

○ Scalars – magnitude only

Vectors and Scalars

A. my velocity (3 m/s)B. my acceleration downhill (30 m/s2)C. my destination (the lab - 100,000 m east)D. my mass (150 kg)

Which of the following cannot be a vector ?

While I conduct my daily run, several quantities describe my condition

Vectors

A vector is composed of a magnitude and a direction Examples: displacement, velocity, acceleration Magnitude of A is designated |A| or A Usually vectors include units (m, m/s, m/s2)

A vector has no particular position

(Note: the position vector reflects displacement from the origin)

Comparing Vectors and Scalars A scalar is an ordinary number. A magnitude without a direction May have units (kg) or be just a number Usually indicated by a regular letter, no

bold face and no arrow on top.Note: the lack of specific designation of a

scalar can lead to confusion

A B

Vectors and their Properties Equality of Two Vectors

Two vectors are equal if they have the same magnitude and the same direction○ Displacement can be the same for many

paths

Movement of vectors in a diagramAny vector can be moved parallel to itself

without being affected

Vectors and their Properties (cont.)

Negative VectorsTwo vectors are negative if they have the same

magnitude but are 180° apart (opposite directions

A = -B A + (-A) = 0 Resultant Vector

The resultant vector is the sum of a given set of vectors

R = C + D


Adding Vectors Geometrically

Scale drawings○ Triangle Method

AlgebraicallyMore convenient

○ Adding components of a vector


Adding Vectors Geometrically

1. Draw the first vector with the appropriate length and in the direction specified

2. Draw the second vector with the appropriate length and direction with its tail at the head of the first vector

3. Construct the resultant (vector sum) by drawing a line from the tail of the first to the head of the second


When you have many vectors, just keep repeating the process until all are included

The resultant is still drawn from the origin of the first vector to the end of the last vector


Vectors obey the Commutative Law of AdditionThe order in which the

vectors are added doesn’t affect the result

A + B = B + A


Vector Subtraction Same method as vector

addition

A – B = A + (-B)


Scalar multiplication Scalar x Vector

Change magnitude or direction of vector

Applies to division also

A

3A

-1A

(1/3)A

Components of a Vector

Vectors can be split into componentsx direction componenty direction component

○ z component too! But we won’t need it this semester!

A vector is completely described by its components

Choose coordinates Rectangular

Components of a Vector (cont.)

Components of a vector are the projection along the axis

Ax = A cos θAy = A sin θ

Then, Ax + Ay = A

Looks like TrigBecause it is!

cos θ = Ax / A


Hypotenuse is A (the vector)Magnitude is defined by Pythagorean

theorem

√(Ax2+ Ay

2)= A

Direction is angle

tan θ = Ay / Ax

Equations valid if θ is respect to x-axis Use components instead of vectors!


Adding Vectors Algebraically1. Draw the vectors2. Find the x and y components of

all the vectors3. Add all the x components4. Add all the y components

If R = A + B,Then Rx = Ax + Bx

and Ry = Ay + By

5. Use Pythagorean Theorem to find magnitude and tangent relation for angle


Example:

A golfer takes two putts to get his ball into the hole once he is on the green. The first ball displaces the ball 6.00 m east, the second 5.40 m south. What displacement would have been needed to get the ball into the hole on the first putt?


Example

A hiker begins a trip by first walking 25.0km southeast from her base camp. She then walks 40.0km, 60.0° north of east where she finds the forest ranger’s tower.

a) Find the components of A

b) Find the components of B

c) Find the components of the resultant vector R = A + B

d) Find the magnitude and direction of R

Displacement, Velocity, and Acceleration in Two Dimensions

So, why all this time to study vectors? We can more describe 2D motion (3D too)

more generally We can apply to it to a variety of physical

problemsForceWorkElectric FieldDisplacement, Velocity, and Acceleration

Motion in Two Dimensions Two dimensional motion under constant

acceleration is known as projectile motion. Projectile path is known as trajectory. Trajectory can fully be described by equations

○ QM – cannot be fully described

Motion in x direction and y direction are independent of each other

Motion in 2D (cont.)

AssumptionsIgnore air resistanceIgnore rotation of the earth (relative)Short range, so that g is constant

Object in 2D motion will follow a parabolic pathBall throw

Projectile MotionImportant points of projectile motion:

1. It can be decomposed as the sum of horizontal and vertical motions.

2. The horizontal and the vertical components are totally independent of each other

3. The gravitational acceleration is perpendicular to the ground so it affects only the perpendicular component of the motion.

4. The horizontal component of the motion has zero acceleration, because the gravitational acceleration has no horizontal component and we neglect the air drag.

Motion in 2D

Falling ballsBoth balls hit the

ground at the same timeInitial horizontal motion

of yellow ball does not affect its vertical motion

Motion in 2D Projectile motion can be decomposed as the

sum of vertical and horizontal components


Projectile Motionx direction uniform

motion○ ax = 0

y direction constant acceleration○ ay = -g

Initial velocity can be broken into componentsv0x = v0cosθ0

v0y = v0sin θ0


Projectile Motion

Complimentary initial angles results in the same range

Maximum range? 45º


x direction motion ax = 0

v0x = v0cosθ0 = vx = constant

Since velocity is constant, a = 0 So the only useful equation is, Δ x = vxot

Δx = v0cosθ0 t

○ From our four equations of motion, this one is only operative equation with uniform velocity (no acceleration)

Motion in 2D (cont.) y direction motion

v0y = v0sin θ0

free fall problem○ a = -g

take the positive direction as upward uniformly accelerated motion, so all the motion

equations from Chapter 2 hold:○ v = v0sin θ0 - gt○ Δy = v0sin θ0 t - ½ gt2

○ v2= (v0sin θ0)2 - 2g Δy Not so concerned about the average velocity in

projectile problems

Concept Test

You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:1) quickly lag behind the plane while falling2) remain vertically under the plane while falling3) move ahead of the plane while falling4) not fall at all

Concept Test

2) remain vertically under the plane while falling

ConcepTest ConcepTest Firing Balls IFiring Balls I

A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

1) it depends on how fast the cart is 1) it depends on how fast the cart is movingmoving

2) it falls behind the cart2) it falls behind the cart

3) it falls in front of the cart3) it falls in front of the cart

4) it falls right back into the cart4) it falls right back into the cart

5) it remains at rest5) it remains at rest

ConcepTest ConcepTest Firing Balls IFiring Balls I

A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

1) it depends on how fast the cart is 1) it depends on how fast the cart is movingmoving

2) it falls behind the cart2) it falls behind the cart

3) it falls in front of the cart3) it falls in front of the cart

4) it falls right back into the cart4) it falls right back into the cart

5) it remains at rest5) it remains at rest

when viewed from

train

when viewed from

ground

In the frame of reference of the cart, the ball only has a verticalvertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocitysame horizontal velocity, so the ball still returns into the cart. http://www.youtube.com/watch?v=FLUOgO2-0lA

Movie

Monkey and the HunterWill the ball go over or under the monkey?http://www.youtube.com/watch?v=cxvsHNR

XLjwWhat if I adjust the speed?http://

www.youtube.com/watch?v=nwQwk15TAh4


Example A stone is thrown upward from the

top of a building at an angle of 30.0° to the horizontal and with an initial speed of 20.0 m/s. The point of release is 45.0m above the ground.

a) Find the time of flight

b) Find the speed at impact

c) Find the horizontal range of the stone


Example:

An artillery shell is fired with an initial velocity of 300 m/s at 55.0° above the horizontal. To clear the avalanche, it explodes on the mountainside 42.0 s after firing. What are the x- and y-coordinates of the shell where it explodes, relative to its firing point?


Summary of Projectile MotionAssuming no air resistance, x-direction

velocity is constanty-direction is similar to free fall problem

○ velocity, displacementx-direction and y-direction are independent

of each other

Chapter 3 Vectors and Two Dimensional Motion. Vectors Motion in 1D Negative/Positive for Direction ○ Displacement ○ Velocity Motion in 2D and 3D 2.

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