Two-Dimensional Motion and Vectors

3.1 Introduction to Vectors

Scalars & Vectors Scalar quantities have magnitude only Speed, volume, # students Vectors have magnitude & direction Velocity, force, weight, displacement

Representing Vectors Boldface type: v is a vector; v is not Symbol with arrow: Arrow drawn to scale: -----> 5m/s, 0° (east) ----------> 10 m/s, 0° (east) <--------------- 15 m/s, 180° (west)

Resultant Vectors Are the sum of two or more vectors Are “net” vectors Can be determined by various methods Graphical addition Mathematically Pythagorean theorem Trigonometry

Graphical Addition of Vectors Parallelogram method Head-to-tail method Draw vectors to scale, with direction, in head-to-tail fashion Resultant drawn from tail of first vector to head of last vector Measure length and angle to determine magnitude and direction

Head-to-Tail Addition of Vectors

Head-to-Tail Addition of Vectors

Other Properties of Vectors Vectors may be moved parallel to themselves when constructing vector diagrams Vectors may be added in any order Vectors are subtracted by adding its opposite Vectors multiplied by scalars are still vectors

Vectors may be added in any order

Direction of Vectors Degrees are measured counterclockwise from x-axis

Direction of Vectors Direction may be described with reference to N, S, E, or West axis 30° West of North 60° North of West Or 120°

3.2 Vector Operations Adding parallel vectors Simple arithmetic

Perpendicular Vectors & the Pythagorean Theorem If two vectors are perpendicular, then you can use the Pythagorean theorem to determine magnitude only Example: if Δx = 40 km/h East Δy = 100 km/h North Then d = (402 + 1002)½ d ≈ 108 km/h But what is the direction?

Basic Trig Functions hyperphysics.phy-astr.gsu.edu

Tangent Function When two vectors are perpendicular: Use Pythagorean theorem to find the magnitude of the resultant vector Use the tangent function to find the direction of the resultant vector

Sample Problem Indiana Jones climbs a square pyramid that is 136 m tall. The base is 2.30 x 102 m wide. What was the displacement of the archeologist? What is the angle of the pyramid

Resolving Vectors Vectors can be “resolved” into x- and y- components Resolve = Decompose = Break down Trig functions are used to resolve vectors

Resolving Vectors into X & Y Components For the vector A Horizontal component = Ax = A·cos θ Vertical component = Ay = A·sin θ hyperphysics.phy-astr.gsu.edu

Resolving a Vector A helicopter travels 95 km/h @ 35º above horizontal. Find the x- and y- components of its velocity.

Adding Non-perpendicular Vectors A + B = R

Resolving Vectors into x and y Components hyperphysics.phy-astr.gsu.edu

Adding vectors that are not perpendicular Two or more vectors can be added by decomposing each vector Add all x components to determine Rx Add all y components to determine Ry Determine magnitude of the resultant R using Pythagorean theorem Determine direction angle θ of resultant using tan-1

Decomposition of Vectors Vector Analysis Vector Analysis Worksheet Magnitude (units) Direction (deg) Decomposition of Vectors Vector A 6 25 Ax = 5.44 Ay = 2.54 Vector B 3 60 Bx = 1.50 By = 2.60 Vector C Cx = 0.00 Cy = Vector D Dx = Dy = Resultant 8.6 36.5 Rx = 6.94 Ry = 5.13

3.3 Projectile Motion phet.colorado.edu Objectives Recognize examples of projectile motion Describe the path of a projectile as a parabola Resolve vectors into components and apply kinematic equations to solve projectile motion problems

Projectile Motion Motion of objects moving in two dimensions under the influence of gravity Baseball, arrow, rocket, jumping frog, etc. Projectile trajectory is a parabola

Projectile motion Assumptions of our problems Horizontal velocity is constant, i.e. Air resistance is ignored Projectile motion is free fall with a horizontal velocity

Motion of a projectile Equations relating to vertical motion: ∆y = vyi(∆t) + ½ag(∆t)2 vyf = vyi + ag∆t vyf2 = vyi2 + 2ag∆y Equations relating to horizontal motion: ∆x = vx∆t vx = vxi = constant (an assumption relating to Newton’s 1st law of motion)

Horizontal Projectile vx = 5.0 m/s t (s) vy vx Δy Δx vy = agt   vx = 1 Δy = 1/2agt2 2 Δx = vxΔt 3 4 5

Horizontal Launch Comparison of vx & vy vectors

Driving off a Cliff A stunt driver on a motorcycle speeds horizontally of a 50.0m high cliff. How fast must the motorcycle leave the cliff in order to land on level ground below, 90.0m from the base of the cliff? Ignore air resistance. Sketch the problem List knowns & unknowns Apply relevant equations

Driving off a Cliff Known: ∆x = 90.0m; ∆y = -50.0m; ax = 0; ay = -g = -9.81 m/s2; vyi = 0 Unknown: vx; ∆t Strategy: vx = ∆x/∆t Since ∆tx must = ∆ty determine ∆t from the vertical drop Now solve for vx using ∆t

Effect of Gravity on Ballistic Launch [physicsclassroom.com]

Projectile Motion: Horizontal vs Angled Horizontal Launch Ballistic Launch www.ngsir.netfirms.com/englishhtm/ThrowABall.htm

Sample Projectile MotionProblem A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º. How long will it be in the air? How high will it go? How far will it go?

Sample Problem A ball is thrown with an initial velocity of 50.0 m/s at an angle of 60º. How long will it be in the air? Known: vi = 50.0 m/s, θ = 60º, a = -g = -9.81 m/s2; Find: Δt, total time in the air

Sample Problem A ball is thrown with an initial velocity of 50 m/s at an angle of 60. How high will it go?

Sample Problem A ball is thrown with an initial velocity of 50 m/s at an angle of 60º. How far will it go?

3.4 Relative Motion Objectives Describe motion in terms of frames of reference Solve problems involving relative velocity

Frames of Reference Motion is relative to frame of reference To an observer in the plane, the ball drops straight down (vx = 0) To an observer on the ground, the ball follows a parabolic projectile path (vx ≠ 0) Frame of reference: a coordinate (defined by the observer) system for specifying the precise location of objects in space A frame of reference is a “point of view” from which motion is described

Relative Velocity Relative velocity of one object to another is determined from the velocities of each object relative to another frame of reference

Example See problem 1, page 109 Use subscripts to indicate relative velocities vbe = vbt + vte vbe = -15 m/s + 15 m/s vbe = 0 m/s

Example Car A travels 40 mi/h north; Car B travels 30 mi/h south. What is the velocity of Car A relative to Car B? vae = 40; vbe = -30; veb = +30 Find vab vab = vae + veb vab = 40 + 30 vab = 70 mi/h North

Relative Velocity One car travels 90 km/h north, another travels 80 km/h north. What is the speed of the fast car relative to the slow car? vfe = 90 km/h north; vse = 80 km/h north vfs = vfe + ves vfs = 90 + (-)80 vfs = 10 km/h north