Chapter 3: Vectors & 2D...

Chapter 3: Vectors & 2D

Motion Brent Royuk

Phys-111 Concordia University

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Vectors •  What is a vector? •  Examples? •  Notation:

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! 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?