PHYSICS 231 Topic 1: 2D Motion

Current Assignments •  Homework Set 1 due this Thursday, Jan

20, 11 pm •  Homework Set 2 due Thursday, Jan 27,

11pm •  Reading: Chapter 4,5 for next week

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Key Concepts: 2D motion •  Vector. Displacement, Velocity, and Acceleration

are vectors. •  Representations: Graphical, angle and magnitude,

components. •  Add, subtract vectors. Decompose in components,

and add components to obtain total vector. •  Treat 2D motion as 2 separate problems, one for

x-direction and one for y-direction. (decompose all vectors in x- and y- components)

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Clicker Quiz

•  What is a vector A: A set of coordinates B: A quantity that has a magnitude and a

direction C: A pointer that indicates a direction D: An arrow

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Clicker Quiz

•  If A is a vector, Ax denotes A: The x-component B: The magnitude in x-direction C: The length D: The magnitude of the vector.

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Vector Concept •  A vector is a quantity that has a direction

and a magnitude •  Examples so far: displacement, velocity,

and acceleration •  In 1D the direction was given by the sign

and a coordinate axis direction. •  In 2D its more complicated. Different

representations can be used

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Representation: Graphical •  Graphically as arrow:

– Direction is direction of arrow – Magnitude is length of arrow (need to define

some scale)

•  Note: where one draws the arrow is irrelevant.

•  Addition by adding arrows (form a chain) •  Subtraction by adding a inverted arrow

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Clicker Quiz

•  Which 2 vectors are the same?

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A B C

D: They are all different vectors

Vector addition/subtraction

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River, water velocity:

Velocity of boat relative to the water vr

vb Vr + Vb

Which of the vectors below best represents the vector sum P + Q?

Checking Understanding → →

Slide 3-16

Checking Understanding Which of the vectors below best represents the difference P – Q?

→ →

Slide 3-18

Checking Understanding Which of the vectors below best represents the difference Q – P?

Slide 3-20

→ →

Representation: Components •  Components: A = Ax + Ay

Magnitudes Ax and Ay are called x- and y-components

•  Calculate components for given A

Ax=A cosΘ, Ay=A sinΘ •  Calculate magnitude of A if components are given:

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Ax

Ay θ A

X

y Parallel to y-axis

Parallel to x-axis

What are the x- and y-components of these vectors?

A.  3, 2 B.  2, 3 C.  -3, 2 D.  2, -3 E.  -3, -2

Checking Understanding

Slide 3-23

What are the x- and y-components of these vectors?

A.  3, 2 B.  2, 3 C.  3, 4 D.  1, 1 E.  4, 4

Checking Understanding

Slide 3-23

What are the x- and y-components of these vectors?

A.  3, -1 B.  3, -4 C.  -3, 3 D.  4, -3 E.  -3, -4

Checking Understanding

Slide 3-25

2D Motion

•  Treat as two separate 1D problems with time as the common parameter

•  In X-direction (calculate with x-components):

•  In Y-direction (calculate with y-components):

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€

vx (t) = v0x + axt

€

x(t) = x0 + v0xt +12axt

2

€

vy (t) = v0y + ayt

€

y(t) = y0 + v0yt +12ayt

2

Vector algebra graphical •  Addition: A + B=C

add beginning of B to end of A, C points then from beginning of A to end of B

•  Negative vector: B=-A: B has opposite direction of A, but same magnitude (flip arrow)

•  Subtraction: A – B = C is same as A + (-B) = C (flip direction of B then add)

•  Multiply with scalar s: B = A * s direction stays the same, multiply length with s (for s<0 flip direction)

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Vector algebra components

•  In component notation things become simple:

•  Addition: add components •  Subtraction: subtract components •  Multiply with scalar: multiply each

component

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Clicker Quiz

•  A cart is moving along a track when a ball is vertically launched off the moving cart. Will the ball:

A: hit the track behind the cart B: hit the cart C: hit the track ahead of the cart

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Clicker quiz

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•  A spring mechanism on a moving (to the right) cart gives a ball an upward velocity. Which best represents the vector of the initial velocity of the ball just when it leaves the spring?

Horizontal x

Vertical y

Horizontal x

Vertical y

Horizontal x

Vertical y

Horizontal x

Vertical y

A B C D

Clicker Quiz •  The gun barrel is straight and aligned with

the aim of the gun. The monkey is hanging on the tree but will let go exactly at the moment the bullet leaves the gun. Where does the hunter need to aim the gun at in order to hit the monkey?

A: A bit above the monkey B: Right at the monkey C: A bit below the monkey

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