Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Physics I 95.141 LECTURE 3 9/14/09

Department of Physics and Applied Physics95.141, F2009, Lecture 2

Physics I95.141

LECTURE 39/14/09


Clicker Test

• Today’s lecture could possibly go worse than Wednesday’s?– True/False


Outline

• Freely Falling Body Problems• Vectors and Scalars• Addition of vectors (Graphical)• Adding Vectors by Components• Unit Vectors

• What Do We Know?– Units/Measurement/Estimation– Displacement/Distance– Velocity (avg. & inst.), speed– Acceleration


Review of Lecture 2

• Last week we discussed how to describe the position and motion of an object

• Reference Frames• Position• Velocity• Acceleration• Constant Acceleration

2

)(2

2

1

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv


LP-HW 2.65

• A stone is dropped from a cliff and the splash is heard 3.4s later. If the speed of sound is 340m/s, how high is the cliff?

• Two part problem!

• Draw a diagram!• Coordinate system • Knowns/Unknowns

A) B)


LP-HW 2.65• Choose equations

A) B)


Example Problem II• Batman launches his grappling bat-hook upwards, if the beam it

attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance)

1) Choose coordinate system2) Knowns and unknowns

3) Choose equation(s)


Example Problem II• Batman launches his grappling bat-hook upwards, if the beam it

attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance)

3) Choose equation(s)

4) Solve

2

)(2

2

1

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv


Vectors and Scalars

• A quantity that has both direction and magnitude, is known as a vector.– Velocity, acceleration, displacement, Force,

momentum– In text, we represent vector quantities as

• Quantities with no direction associated with them are known as scalars– Speed, temperature, mass, time

r,a,v


Vectors and Scalars

• In the previous chapter we dealt with motion in a straight line– For horizontal motion (+/- x)– For vertical motion (+/- y)

• Velocity, displacement, acceleration were still vectors, but direction was indicated by the sign (+/-).

• We will first understand how to work with vectors graphically


Addition of vectors

• In one dimension– If the vectors are in the same direction

– But if the vectors are in the opposite direction


Addition of Vectors (2D)

• In two dimensions, things are more complicated


Addition of Vectors

• “Tip to tail” method– Draw first vector– Draw second vector, placing tail at tip of first vector– Arrow from tail of 1st vector to tip of 2nd vector is

)(2

)(3

2

1

ymD

xmD


Commutative property of vectors

• “Tip to tail” method works in either order– Draw first vector– Draw second vector, placing tail at tip of first vector– Arrow from tail of 1st vector to tip of 2nd vector is

)(2

)(3

2

1

ymD

xmD

1221 DDDD


Three or more vectors

• Can use “tip to tail” for more than 2 vectors

+ + =


Subtraction of vectors

• For a given vector the negative of the vector is a vector with the same magnitude in the opposite direction.

1V

1V

)( 2121 VVVV

- = +

• Difference between two vectors is equal to the sum of the first vector and the negative of the second vector


Multiplying a vector by a scalar

• You can also multiply a vector by a scalar

• When you do this, you don’t change the direction of the vector, only its magnitude

1Vc

c=2 c=4 c=-2


Adding vectors by components

• Adding vectors graphically is useful to understand the concept of vectors, but it is inherently slow (not to mention next to impossible in 3D!!)

• Any 2D vector can be decomposed into components


Determining vector components

• So in 2D, we can always write any vector as the sum of a vector in the x-direction, and one in the y-direction.

• Given V(V,θ), we can find Vx and Vy

yx VVV


Example

• A vector is given by its magnitude and direction:

• Write the vector in terms of its components

axisxabovemV

45,10

mV

mV

y

x

07.7

07.7


Determining vector components

• Or, given Vx and Vy, we can find V(V,θ).

x

y

yx

V

V

VVV

tan

22


Example

• A vector is given by its vector components:

• Write the vector in terms of magnitude and direction

4,2 yx VV

axisxfrom

axisxfrom

V

117

63

47.420


Adding vectors by components

• Given V1 and V2, how can we find V= V1 + V2?

yx

yyxx

VV

VVVV

VVV

2121

21

V1

V2V


Example

• A trucker drives 10miles North, then 40 miles 30º East from South. What is his final displacement?

• Set up coordinate system


Example


• Resolve each vector into x,y components


Example


• Add vector componentsmilesVmilesV

milesVmilesV

yx

yx

6.34,20

10,0

22

11

milesVmilesV yx 6.24,20


3D Vectors

• Adding vectors vectors by components is especially helpful for 3D vectors.

• Also, much easier for subtraction

zzyyxx

zyxzyx

VVVVVVVVV

VVVVVVVV

21212121

22221111 ,

zzyyxx

zyxzyx

VVVVVVVVV

VVVVVVVV

21212121

22221111 ,


3D Vectors

• Which is traveling fastest?

sm

sm

zsm

ysm

x

sm

zsm

ysm

x

VV

VVVV

VVVV

220

50,200,100

50,200,10

3

2222

1111

• Three cars are traveling with velocities given by:


Unit Vectors

• Up to this point, we have written vectors in terms of their components as follows:

• There is an easier way to do this, and this is how we will write vectors for the remainder of the course:

vectorsunitasknownkji

kVjViVV zyx

ˆ,ˆ,ˆ

ˆˆˆ


Unit Vectors

• What are unit vectors?– Unit vectors have a magnitude of 1 and point along major axes

of our coordinate system

• Writing a vector with unit vectors is equivalent to multiplying each unit vector by a scalar

kVjViVV

kVVjVViVV

zyx

zzyyxx

ˆˆˆ

ˆ,ˆ,ˆ


Unit Vectors

• For a vector with components:

• Write this is unit vector notation

2,3,4 zyx VVV


Example: Vector Addition/Subtraction• Displacement

– A hiker traces her movement along a trail. The first leg of her hike brings her to the foot of the mountain:

– On the second leg, she ascends the mountain, which she figures to be a displacement of:

– On the third, she walks along a plateau.

– Then she falls of a cliff

– What is the hiker’s final displacement?

jmimV ˆ500ˆ25001

kmjmimV ˆ700ˆ700ˆ5002

jmV ˆ6003

kmV ˆ5004


Example: Vector Addition/Subtraction

jmimV ˆ500ˆ25001

kmjmimV ˆ700ˆ700ˆ5002

jmV ˆ6003

kmV ˆ5004

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Department of Physics and Applied Physics 95.141, F2009, Lecture 2 Physics I 95.141 LECTURE 3 9/14/09