Unit 2: Motion in 2D Textbook: Chapter 3

Unit Objectives: Motion Models 1. Determine which model (constant velocity or constant acceleration, or varying acceleration) is appropriate to describe the horizontal and vertical component of motion of an object

Unit Objectives: Projectiles 3. Use appropriate kinematic equations to determine the position or velocity of a projectile at a specific point. Sketch the graph of motion for projectiles a) y-x, y-t, x-t, v x – t, v y - t, a x -t, a y -t 4. Given information about the initial velocity and height of a projectile, determine a) time of flight, b) the point where a projectile lands, and c) velocity at impact

Unit Objectives: Vectors 5. Graphical representation of Vectors a) Given a vector, draw its components b) Recognize the magnitude and direction of a vector from a vector diagram c) Determine the sum of 2 or more vectors graphically 6. Numerical Analysis of Vectors a) Given the magnitude and direction of a vector, determine the components using trig b) Given the components of a vector, determine the magnitude and direction using Pythagorean Theorem and trig c) Determine the sum of 2 or more vectors using Pythagorean Theorem and trig d) Represent by using unit vectors i, j, & k.

Unit Objectives: Relative Motion 7. Use vectors to perform relative velocity calculations

Vectors: How much & which way? When describing motion, often the questions asked are “How far?” or “How fast?” However, for a person that is lost, “which way?” becomes more valuable. Vectors answer both questions: 1 – How much (magnitude)? 2 – Which way (direction)?

Vectors & Scalars Scalars have magnitude only Quantity of something Distance, speed, time, mass, temperature Vectors have both magnitude and direction displacement, velocity, acceleration R  head tail

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. Can also be given by using unit vectors A x 

Magnitude of Vectors The magnitude of a vector is the size of whatever the vector represents and is represented by the length of the vector drawn. Symbolically, the magnitude is often represented as │A │ A If vector A represents a displacement of three kilometers to the north… B Then vector B, which is twice as long, would represent a displacement of six kilometers to the north!

Polar Notation Magnitude and direction of the vector are stated separately. Magnitude is a positive number and the angle is made with the positive x-axis v = 5 m/s at 135˚

Rectangular Notation Defining a vector by its components y-component: vector projection parallel to y-axis x-component: vector projection parallel to x-axis VxVx VyVy

Converting Polar & Rectangular

A B R A + B = R Graphical Addition of Vectors Vectors are added graphically together tail-to-tip. The sum is called the resultant. The inverse, is called the equilibrant.

Graphical Addition of Vectors

Vector Subtraction Define the negative of a vector (inverse) Has the same magnitude but in the opposite direction Add the negative Vector

Vector Multiplication by a Scalar A vector can be multiplied by a scalar c The result is a vector c Same direction but a magnitude of cV If c is negative, the resultant vector points in the opposite direction.

Vector Addition by Components Any vector can be expressed as the sum of two other vectors called components. Components are chosen so they are perpendicular to each other. Can be found using trigonometric functions

Vector Addition By Components The components are effectively one-dimensional, so they can be added arithmetically.

Vector Addition By Components 1) Resolve each vector into its x- and y-components. A x = A cos  A y = A sin  B x = B cos  B y = B sin  2) Add the x-components together to get R x and the y- components to get R y. 3) Use the Pythagorean Theorem to get the magnitude of the resultant. 4) Use the inverse tangent function to get the angle.

Sample Problem What is the value of “a” and “b”? a = -3 & b = 10

Sample Problem An airplane trip involves three legs with two layovers. Find the total displacement of the plane. 1 st leg: is 620 km due east 2 nd leg: 440 km due southeast 3 rd leg: 550 km 53 ⁰ south of west D R = ⁰ South of East

Sample Problem Add together the following graphically and by components, giving the magnitude and direction of the resultant and of the equilibrant. Vector A: o Vector B: o Vector C: o Resultant: o Equilibrant: o

Yet another sample!!! A jogger breaks her workout into three segments: jogging, sprinting, and walking. Starting at home, she jobs a displacement vector of (a, 2a) blocks, sprints a displacement of (3b, b) blocks, and walks back home with a displacement of (2, -6) blocks. What is the vector value of her displacement during the sprint? Sprint (-6, -2) blocks

Consider Three Dimensions Easier to represent using unit vector notation!!

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. z y x i j k

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 j and B = i j. Calculate C where C = A + B. C = i j

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). 5 i + 10 j + 3 k

Suppose I need to convert unit vectors to a magnitude and direction? Given the vector

Back to 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? m

1 Dimension2 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 = v x i + v y j + v z k a = a x i + a y j + a z k In Unit Vector Notation

Sample Problem The position of a particle is given by r = (80 + 2t)i – 40j - 5t 2 k. Derive the velocity and acceleration vectors for this particle. What does motion “look like”? v = 2 i - 10t k a = -10 k freefall

Another Sample A position function has the form r = x i + y j with x = t 3 – 6 and y = 5t - 3. What are the velocity and acceleration functions? What are the velocity and acceleration at t=2s? v = 3t 2 i + 5 jv(2) = 12 i + 5 j a = 6t ia(2) = 12 i

Practice Problems 1- A baseball outfielder throws a long ball. The components of the position are x = (30 t) m and y = (10 t – 4.9t 2 ) m Write vector expressions for the ball’s position, velocity, and acceleration as functions of time. Use unit vector notation! Write vector expressions for the ball’s position, velocity, and acceleration at 2.0 seconds. 2- 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?

Projectiles An object that moves in two dimensions under the influence of only gravity Accomplished by usually launching at an angle or going off a flat surface with initial horizontal velocity. Neglect air resistance Follow parabolic trajectory

Launch Angle The components v ix & v iy are not necessarily positive. If an object is thrown downward, then v iy is negative.

Projectiles & Acceleration If you take an object and drop, it will fall straight down and not sideways a x = 0 & a y =g = -9.8 m/s 2 The vertical component of acceleration is just the familiar g of free fall while the horizontal is zero

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/s 2 for the entire trajectory.

Trajectory of Projectile vxvx vyvy vyvy vxvx vxvx vyvy vxvx vyvy vxvx The velocity can be resolved into components all along its path. Horizontal velocity remains constant; vertical velocity is accelerated.

Trajectory Path of a Projectile

Position graphs for 2-D projectiles. Assume projectile fired over level ground. x y t y t x

Acceleration graphs for 2-D projectiles. Assume projectile fired over level ground. t ayay t axax

Lets think about this!!! A heavy ball is thrown exactly horizontally at height h above a horizontal field. At the exact instant that ball is thrown, a second ball is simply dropped from height h. Which ball hits firsts? (demo-x-y shooter)

Two Independent Motions 1) Uniform motion at constant velocity in the horizontal direction 2) Free-Fall motion in the vertical direction

Remember… To work projectile problems… …resolve the initial velocity into components.  VoVo V o,y = V o sin  V o,x = V o cos 

Practice Problems 1) A soccer player kicks a ball at 15 m/s at an angle of 35 o above the horizontal over level ground. How far horizontally will the ball travel until it strikes the ground? 2) A cannon is fired at 100m/s at an 15 o 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? 3. Students at an engineering contest use a compressed air cannon to shoot a softball at a box being hoisted straight up at 10 m/s by a crane. The cannon, tilted upward at 30 degree angle, is 100 m from the box and fires by remote control the instant the box leaves the ground. Students can control the launch speed of the softball by setting air pressure. What launch speed should the students use to hit the box?

Range Equation Derive the range equation for a projectile fired over level ground.

Acceleration in 2-D A runner is going around a track. She is initially moving with a velocity vector of (0.00, -8.00) m/s and her constant acceleration is (1.10, 1.10) m/s 2. What is her velocity 7.23 seconds later. Round the final velocity components to the nearest 0.01 m/s.

Multidimensional Motion - Calculus What is the velocity function of the plane? What is the velocity at t = 2 seconds? Just like in 1-D, take the derivative of the position function, to get the velocity function. Take the double derivative to find acelleration…

Unit Vectors & Calculus Treat the same way as you do with one dimensional motion Take the derivative or integral for each unit vector

Reference Frames Coordinate system used to make observations. The woman is using the surface of the Earth as her reference frame. She considers herself and the train platform to be stationary, while the train is moving to the right with positive velocity.

Reference Frames cont. If now, the perception of motion is from Ted’s point of view (man in the train). He uses the inside of the train as his reference frame. He sees other people in the train as stationary and objects outside the train moving back with a negative velocity.

Reference Frames There is no right or wrong reference frame. Must be clear about which reference frame is being used to assess motion.

Reference Frame Conditions 1. The frames are oriented the same, with the x and y axes parallel to each other 2. The origins of frame A & B coincide at t=0. 3. All motion is in the xy-plane, so we don’t need to consider the z-axis 4. The relative velocity (of the frames) is constant. (a = 0) Inertial Reference Frames

Classical Mechanics are only valid in inertial reference frames. In other words, all observers would measure the same acceleration for a moving body. We will discuss this in more detail when we talk about Newton’s Laws of Motion

Relative Velocity When dealing with relative velocities, must use subscripts to identity the object in motion and the frame of reference. Each velocity is labeled first with the object in motion and second with the reference frame in which it has this velocity. If the reference frame is switched with the object, then the velocity becomes negative.

Relative Velocity v = 15 m/s

Another Sample

Practice Problems V BS = 3.35 m/s at 63.4 degrees

Headwinds, Tailwinds, Crosswinds… Direction of wind is determine from the place of origin Tailwind – wind is coming from the “tail” rear of the plane Wind in direction of motion – helps the motion Headwind – wind is coming from the “head” or front of the plane. Wind direction is opposite to the motion – resists motion Crosswind – wind is coming perpendicular to the plane Wind will steer plane in other direction

Police Car Chasing A motorist traveling west at 77.5 km/h is being chased by a police car traveling at 96.5 km/h. What is the velocity of the motorist relative to the police car?