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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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
Vectors and their Properties (cont.)
Adding Vectors Geometrically
Scale drawings○ Triangle Method
AlgebraicallyMore convenient
○ Adding components of a vector
Vectors and their Properties (cont.)
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
Vectors and their Properties (cont.)
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 and their Properties (cont.)
Vectors obey the Commutative Law of AdditionThe order in which the
vectors are added doesn’t affect the result
A + B = B + A
Vectors and their Properties (cont.)
Vector Subtraction Same method as vector
addition
A – B = A + (-B)
Vectors and their Properties (cont.)
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
Components of a Vector (cont.)
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!
Components of a Vector (cont.)
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
Components of a Vector (cont.)
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?
Components of a Vector (cont.)
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 (cont.)
Projectile Motionx direction uniform
motion○ ax = 0
y direction constant acceleration○ ay = -g
Initial velocity can be broken into componentsv0x = v0cosθ0
v0y = v0sin θ0
Motion in 2D (cont.)
Projectile Motion
Complimentary initial angles results in the same range
Maximum range? 45º
Motion in 2D (cont.)
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
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
Motion in 2D (cont.)
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
Motion in 2D (cont.)
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?
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