chapter 3: vectors & 2d motionestrada.cune.edu/facweb/brent.royuk/phys111/docs/chapter03_3.pdf12...
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![Page 1: Chapter 3: Vectors & 2D Motionestrada.cune.edu/facweb/brent.royuk/phys111/docs/chapter03_3.pdf12 37 Addition of Velocities • What if velocity is not in the direction of x or y axes?](https://reader033.vdocuments.us/reader033/viewer/2022042304/5ecfdf60fde6dd0c864f354c/html5/thumbnails/1.jpg)
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Chapter 3: Vectors & 2D
Motion Brent Royuk
Phys-111 Concordia University
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Vectors • What is a vector? • Examples? • Notation:
€
! a or
! a or a
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Vector Addition • Graphical Methods
– Triangle, parallelogram, polygon
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Graphical Vector Addition • Resultant • Construction vs. Analytical • Right vs. Oblique • Vector mobility • Physical Diagrams vs. Vector
Diagrams • Vector subtraction
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Trig Review
• Remember your trigonometry?
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Trig Review
• For right triangles:
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sin A =ac
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cos A =bc
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tan A =ab
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c2 = a2 + b2
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Vector Addition • Magnitudes have units • Directions have angles • Directional systems
– Heading, 30o N of W – Bearing, Degrees clockwise from N – Cartesian, 20o below the x-axis
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Vector Addition • Directional systems
– Compass, heading, NW, SSE, etc.
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Graphical Addition Examples • If you travel north for 12 km and then
west 8.0 km, what is the magnitude and heading of your displacement?
• A plane flying with a velocity of 120 m/s due south experiences a crosswind of velocity 38 m/s west. What is the plane’s resultant velocity?
• Add these force vectors: 8.0 N at 20� N of E and 10.0 N at 20� W of N.
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Vector Components • Resolving a vector into components
– Expressing vectors in terms of carefully chosen orthogonal vector components
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Vector Components
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! A =
! A x +
! A y
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Ax = ?Ay = ?
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Component Vector Addition • Vector addition made easy (or at least algorithmic) 1. Find x-components for vectors A and B
1. Ax = A cos θ 2. Bx = B cos θ
2. Find y-components for A and B 1. Ay = A sin θ 2. By = B sin θ
3. Find components for the resultant R 1. Rx = Ax + Bx 2. Ry = Ay + By
4. Find the magnitude and direction of R 1. θ= tan-1 (Ry/Rx) 2. R = R
x2 + R
y2
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Component Vector Addition • Pictorial Representation
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Component Addition Examples • A plane leaves an airport and is
later sighted 215 km away, at 22o E of N. How far east and north is the plane from the base?
• Add these forces: 58 N at 60 o W of S and 67 N at 15 o E of N. Give answer as components and also as magnitude and direction.
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Projectile Motion • What path does the ball follow when
dropped?
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Projectile Motion • Horizontal Launch
– What happens if you kick a ball off a cliff?
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Projectile Motion • Falling Comparison
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Projectile Motion • Horizontal Launch
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100-ft Cliffdiving
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Projectile Motion • Velocity Changes
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Projectile Motion
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Projectile Motion • Launch at an angle
– What is the range? – A useful identity: sin 2� = 2 sin�
cos�
v �
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Projectile Motion
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R =v2 sin2θ
g
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Projectile Motion • Air Resistance
– Data: 100 mph at 60o; vacuum = 581 ft., air = 323 ft.
• How about the moon?
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ConcepTest • A battleship simultaneously fires two shells at enemy ships.
If the shells follow the parabolic trajectories shown below, which ship gets hit first?
1. A 2. both at the same time 3. B 4. need more information
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Projectile Motion
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Projectile Motion • a
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Projectile Motion Examples • Find the range of a projectile launched
with vo = 35 m/s at 52� with the ground. How high does it rise? – With what velocity does it land? At what
angle? • An artillery shell with a muzzle velocity
of 125 m/s is fired at an angle of 35.0� with the horizon. If the shell explodes 10.0 s after being fired, where does the blast occur?
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Projectile Motion Examples • A police officer is chasing a burglar across a
rooftop. Both are running at 4.5 m/s. When the burglar reaches the end of the roof he jumps horizontally toward the next building, which is 6.2 m away but 4.8 m lower. Should the policeman jump to pursue or take the elevator to clean up the mess? Justify your answer.
• A plane flies at a velocity of 85.2 m/s at 43� W of N. The velocity of the wind is 24.3 m/s at 18� S of E. Find a) the velocity of the plane in component form and b) the distance the plane travels in 2.5 hours.
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Relative Velocities • Frames of reference • Everyday examples of velocity addition
– baseball from pickup – closing velocity of cars – moving walkway in airport
• Example – Consider a 500 m wide river with flow rate of 0.85
m/s. The boat can travel at rate of 2.3 m/s and is steered directly across the river. Find a) v of boat relative to observer on shore, b) distance traveled downstream while boat crosses, c) total actual distance traveled.
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Relative Velocities
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Relative Velocities
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Relative Velocities
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Addition of Velocities • What if velocity is not in the direction of
x or y axes? – Compare: v, vx, vy, and v – vx = ?, vy = ? – x = vx t, y = vy t, etc.
• Example – An object travels at an angle of 32� with
the x-axis at a speed of 3.2 m/s. In 2.0 seconds, how far does it travel a) in the x-direction, b) in the y-direction and c) total.
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Addition of Velocities • Example
– A plane flies due west toward a destination 600 km away. The plane can fly at 200 km/hr and it points straight west and flies that fast. The total time is 3 hours, right? No: plane experiences a headwind at 24� S of E with speed 23 km/hr. How far away from the destination is the plane after 3 hours?