Chapter 3 Vectors and Two Dimensional Motion

Vectors  Motion in 1D Negative/Positive for Direction ○ Displacement ○ Velocity  Motion in 2D and 3D 2 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/s 2 )  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 diagram Any vector can be moved parallel to itself without being affected

Vectors and their Properties (cont.)  Negative Vectors Two 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  Algebraically More 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 Addition The 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 components x direction component y 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 A x = A cos θ A y = A sin θ  Then, A x + A y = A  Looks like Trig Because it is! cos θ = A x / A

Components of a Vector (cont.)  Hypotenuse is A (the vector) Magnitude is defined by Pythagorean theorem √(A x 2 + A y 2 )= A Direction is angle tan θ = A y / A x  Equations valid if θ is respect to x-axis  Use components instead of vectors!

Components of a Vector (cont.) Adding Vectors Algebraically 1. Draw the vectors 2. Find the x and y components of all the vectors 3. Add all the x components 4. Add all the y components If R = A + B, Then R x = A x + B x and R y = A y + B y 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 problems Force Work Electric Field Displacement, 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.)  Assumptions Ignore air resistance Ignore rotation of the earth (relative) Short range, so that g is constant  Object in 2D motion will follow a parabolic path Ball throw

Projectile Motion Important 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 balls Both balls hit the ground at the same time Initial 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

Motion in 2D (cont.)  Projectile Motion x direction uniform motion ○ a x = 0 y direction constant acceleration ○ a y = -g  Initial velocity can be broken into components v 0x = v 0 cos θ 0 v 0y = v 0 sin θ 0

Motion in 2D (cont.) Projectile Motion  Complimentary initial angles results in the same range  Maximum range?  45º

Motion in 2D (cont.)  x direction motion a x = 0 v 0x = v 0 cos θ 0 = v x = constant Since velocity is constant, a = 0 So the only useful equation is, Δ x = v xo t Δx = v 0 cosθ 0 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 v 0y = v 0 sin θ 0 free fall problem ○ a = -g take the positive direction as upward uniformly accelerated motion, so all the motion equations from Chapter 2 hold: 1. v = v 0 sin θ 0 - gt 2. Δy = v 0 sin θ 0 t - ½ gt 2 3. v 2 = (v 0 sin θ 0 ) 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 falling 2) remain vertically under the plane while falling 3) move ahead of the plane while falling 4) not fall at all

Concept Test 2) remain vertically under the plane while falling

ConcepTest Firing 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 moving 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest

ConcepTest Firing 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 moving 2) it falls behind the cart 3) it falls in front of the cart 4) it falls right back into the cart 5) it remains at rest when viewed from train when viewed from ground vertical same horizontal velocity In the frame of reference of the cart, the ball only has a vertical 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 velocity, so the ball still returns into the cart.

Movie  Monkey and the Hunter Will the ball go over or under the monkey? XLjw XLjw What if I adjust the speed? TAh4 TAh4

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?

Motion in 2D (cont.)  Summary of Projectile Motion Assuming no air resistance, x-direction velocity is constant y-direction is similar to free fall problem ○ velocity, displacement x-direction and y-direction are independent of each other