PHY211 Lecture 4

Matthew Rudolph

Syracuse University

January 23, 2020

Vectors

Describing vectors

You’ve probably used vectors before without really thinking about it“Walk one block east” – displacement vector“Driving northbound at 100 km/h” – velocity vector“The wind was blowing from the west at 30 km/h” – velocity vector

Anything that points a direction, and has some magnitude is a vectorCan even think of just a direction as a vector of magnitude 1

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Describing vectorsMathematically

It’s easy to draw a picture of an arrow pointing some direction withsome length

Use direction and magnitudeOr components along some axes

∣∣~v∣∣θ

x

y

vx

vy

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Describing vectorsMathematically

It’s easy to draw a picture of an arrow pointing some direction withsome lengthUse direction and magnitude

Or components along some axes

∣∣~v∣∣θ

x

y

vx

vy

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Describing vectorsMathematically

It’s easy to draw a picture of an arrow pointing some direction withsome lengthUse direction and magnitudeOr components along some axes

∣∣~v∣∣θ

x

y

vx

vy

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ConversionsWe’ll have to do this a lot!

Best bet – always draw the relevant triangle with the vector on thehypotenuse

Make it its own picture!

∣∣~v∣∣vy

vx

θ

∣∣~v∣∣≡ vvx = v cos(θ )

vy = v sin(θ )

v =√

v2x +v2

y

tan(θ ) = vy/vx

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Question

For this vector, what is vy?A v cos(φ)

B v sin(φ)

C v tan(φ)

D v/cos(φ)

~v

x

y

φ

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Vector placement

Vectors do not “save” their start positionExample: if you move from point A to point B, then the vector betweenthem is your displacement. If you need to know the starting point, thenyou have additional quantities describing the position of AThis means these are the same vectors:

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Adding graphicallyWhat is ~A+~B?

~A

~B

Move tip-to-tail

~A

~BDraw from start to finish

~A

~B~A+~B

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Adding graphicallyWhat is ~A+~B?

~A

~B

Move tip-to-tail

~A

~B

Draw from start to finish

~A

~B~A+~B

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Adding graphicallyWhat is ~A+~B?

~A

~B

Move tip-to-tail

~A

~BDraw from start to finish

~A

~B~A+~B

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Subtracting graphicallyWhat is ~A−~B?

Two options

Add ~A+(−~B)~A

−~B~A−~B

Draw from ~B to ~A

~A

~B~A−~B

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Subtracting graphicallyWhat is ~A−~B?

Two options

Add ~A+(−~B)~A

−~B~A−~B

Draw from ~B to ~A

~A

~B~A−~B

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Subtraction question

You have the following twovectors, and you want to draw~D−~C. Which result iscorrect?

~C

~D(a)

(b)

(c)

(d)

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Why components?

Drawing is useful to get a picture in your head of what is going on, so itwill help us make our graphs and sketches to solve problemsBut components are easier to use with numbersCan simply add x , y , and z components separately(

~A+~B)

x= Ax +Bx(

~A+~B)

y= Ay +By(

~A+~B)

z= Az +Bz

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Pre-lecture question 1

What form does the trajectory of a particle have if the distance theparticle travels from any point A to point B is equal to the magnitude ofthe displacement from A to B?

A L-shapeB CurvedC Straight line

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Pre-lecture question 2

If an object experiences an acceleration in the y direction, does thex-component of its velocity change?

A YesB No

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Pre-lecture question 3

If an object moves 3 meter along the positive x direction, then turnsand moves 4 meter along the negative y direction, what is themagnitude of its displacement?

A 0 mB 7 mC 5 mD −5 m

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Simultaneous problemswith constant acceleration

Just have two sets of the same equations, the only thing they share istime!

vx(t) = vx ,0 +ax t

x(t) = x0 +vx ,0t +12

ax t2

vy (t) = vy ,0 +ay t

y(t) = y0 +vy ,0t +12

ay t2

This covers a lot of what we will do this semester!

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Unit vector form

Sometimes (in the book) you will see the components written out onone lineThis is done by writing each component as a vector pointing in the x ,y , or z directionThen the sum of these component vectors is the full thingUses the notation of unit vectors

ı̂ ≡ x direction̂ ≡ y direction

k̂ ≡ z direction~r = x ı̂+y ̂+zk̂

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Vector recapWhat do we need to know?

We will use vectors all year to describe physics!You should be able to draw vectors; add and subtract them graphically

Very helpful for setting up problems and checking if your answer makessense!Example: if the velocity arrow you want to solve for points towards −x ,your vx should be negative at the end!

For most problems you have to do componentsBe able to convert magnitude and angle to x and y componentsComponents are often like two separate problems that just share time t

Today we will practice this in context of motion

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Skier example

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Skier math

The figure shows a skier moving with an acceleration of 2.1 m/s2 down aslope of 15◦ at t = 0. With the origin of the coordinate system at the front ofthe lodge, her initial position and velocity are

~r(0) = (75.0 ı̂−50.0 ̂) m

and~v(0) = (4.1 ı̂−1.1 ̂) m/s.

(a) What are the x- and y -components of the skier’s position and velocityas functions of time?(b) What are her position and velocity at t = 10.0s?

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Problem solving stepsKey strategy for the class!

1 Draw a picture – it helps visualize things2 Choose axes – which way is positive? Where is zero?3 When is t = 0?4 For motion problems use the equations of motion5 Translate the question into one about your variables6 Do algebra to solve for the unknowns7 Calculate a numerical answer8 Does your answer make sense?

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Simplifying things

Can you think of an easier way to do the last problem?

Why not define +x in the direction of acceleration?In two dimensions you can pick the angle of your axes!

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Simplifying things

Can you think of an easier way to do the last problem?Why not define +x in the direction of acceleration?In two dimensions you can pick the angle of your axes!

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Example

A spaceship is drifting with a velocity~v = 100m/s ı̂+200m/s ̂ at t = 0. It thenturns on its engine which provides a constantacceleration. Four seconds later thespaceship’s velocity is~v = 100m/s ̂. What isthe spaceship’s acceleration in componentform? What is the magnitude ofacceleration? What is the position as afunction of time?

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Vectors and motion

Once we can move in more thanone dimension, need to worryabout relative direction ofposition, velocity, andaccelerationIn 1D, acceleration had to pointeither with velocity, or oppositeBut in 2 or 3D we can turn

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Does that make sense?

Let’s look at thecomponents of~awith a goodchoice of axis

~v

xy

∣∣~v∣∣= vx

~a

axay

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Does that make sense?

Let’s look at thecomponents of~awith a goodchoice of axis

~v

xy

∣∣~v∣∣= vx

~a

axay

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Does that make sense?

Let’s look at thecomponents of~awith a goodchoice of axis

~v

xy

∣∣~v∣∣= vx

~a

axay

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Does that make sense?

Let’s look at thecomponents of~awith a goodchoice of axis

~v

xy

∣∣~v∣∣= vx

~a

axay

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Radial and tangential

Velocity is always tangential to the path – that’s basically its definition!It tells you where you are going in the next instant of time

Acceleration can have a tangential component – that makes you speedup or slow downAnd a radial component – that makes you turn!

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Radial and tangential

Velocity is always tangential to the path – that’s basically its definition!It tells you where you are going in the next instant of timeAcceleration can have a tangential component – that makes you speedup or slow downAnd a radial component – that makes you turn!

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Reminders

Read sections 4.3 for next Tuesday and do pre-lecture questionsFirst homework due tomorrow at recitation

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