department of physics and applied physics 95.141, f2009, lecture 4 physics i 95.141 lecture 4...

Department of Physics and Applied Physics95.141, F2009, Lecture 4

Physics I95.141

LECTURE 49/16/09


Administrative Notes

• Masteringphysics.com– Feedback welcome!– Keep it civil, respectful

• These are to help you review the previous lecture and prepare for the upcoming.

• They are a tool for me to make sure people understand the material and

• For you to make sure you understand the material

– I recognize that issues like hints and sig figs are causing some confusion.

– Grading focus will be more on attempting the problems and spending time working on them, not necessarily final score.

• Help in class.• HW Review sessions


Example Problem II (Lecture 3)• Batman launches his grappling bat-hook upwards, if the walkway it

attaches to is 25m 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 resistance)

3) Choose bat-equation(s)

4) Solve

2

)(2

2

1

22

2

o

oo

oo

o

vvv

xxavv

attvxx

atvv


Outline

• Vector Kinematics• Relative Motion

• What do we know?– Units/Dimensions/Measurement/SigFigs– Kinematic equations– Freely falling objects– Vectors


Vector Review• If I have 3 vectors given by:

kjiV

kjiV

kjiV

ˆ4ˆ3ˆ3

ˆ3ˆ1ˆ2

ˆ10ˆ2ˆ4

3

2

1

• Which of the following vectors corresponds to321 VVVVR

kjiVd

kjiVc

kjiVb

kiVa

R

R

R

R

ˆ3ˆ6ˆ3)

ˆ11ˆ6ˆ9)

ˆ3ˆ6ˆ9)

ˆ3ˆ3)


Vector Kinematics

• We can now do kinematics in more than one dimension– This is helpful, because we live in a 3D world!

• We previously described displacement as Δx, but this was for 1D, where motion could only be positive or negative.

• In more than 1 dimension, displacement is a vector

v

r


Vector Kinematics12 xxx

12 rrr


Vector Kinematics

• In unit vectors, we can write the displacement vector as:

• We can now rewrite our expression for average velocity:


Vector Kinematics

• Average velocity only tells part of the story• Just like for motion in 1D, we can let Δt get smaller and smaller….• Gives instantaneous velocity vector:

t

r

tv

0

lim


Velocity Vector

• The magnitude of the average velocity vector is NOT equal to the average speed.

• But the magnitude of the instantaneous velocity vector is equal to the instantaneous speed at that time


Instantaneous Velocity (math)

• To find the instantaneous velocity, we can take the derivative of the position vector with respect to time:

kvjvivdt

rdk

dt

dzj

dt

dyi

dt

dx

dt

rdv zyx

ˆˆˆˆˆˆ


Example

• Say we are given the position of an object to be:

ktjeittr t ˆ)2sin(ˆ2ˆ)14()( 2

• Can we find the velocity as a function of time?


Acceleration Vector

• Average acceleration:

• Instantaneous acceleration


Acceleration

• Acceleration will be non-zero not only for a change in speed, but for movement at constant speed but changing direction

kdt

zdj

dt

ydi

dt

xd

kdt

dvj

dt

dvi

dt

dv

dt

vda yyx

ˆˆˆ

ˆˆˆ

2

2

2

2

2

2

• An object moves around a circle at a constant speed. It’s acceleration must then be:– A) 0– B) Constant– C) Constantly changing


Example Problem

• Imagine we are given the position of an object as a function of time– Find displacement at t=1s and t=3s– Find velocity and acceleration as a function of time– Find velocity and acceleration at t=3s

jtmitttrs

ms

ms

m ˆ)(3)(3ˆ)(2)(4)( 3232


Example Problem

• Imagine we are given the position of an object as a function of time– Find displacement at t=1s and t=3s– Find velocity and acceleration as a function of time– Find velocity and acceleration at t=3s

jtita

jtittv

sm

sm

sm

sm

sm

ˆ)(18ˆ)(4)(

ˆ)(9ˆ)(4)(4)(

32

322


Example Problem

• Let’s say we are told that a Force causes an object to accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).

– A) Give the initial velocity vector of the object

– B) Plot x(t) vs. t

– C) Plot y(t) vs. t

– D) Plot the object’s trajectory in the xy plane


Example Problem• Let’s say we are told that a Force causes an object to

accelerate in the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0)..– A) Give the initial velocity vector of the object

20

20

vx

vy

3.56,18 smv


Example Problem• Let’s say we are told that a Force causes an object to accelerate in

the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).– Before we solve B-D, let’s determine equations of motion

(METHOD I)



the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).– Before we solve B-D, let’s determine equations of motion

(METHOD II)



the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, and starts at the point (0,0).– B) Plot x(t) vs t

10

100

t(s)

x(t)

time x(t)

0 0m

1 10m

2 20m

5 50m

10 100m

ttvtx ox 10)(



the -y direction at 5m/s2. The object has an initial velocity in the +x direction of 10m/s, and in the +y direction of 15 m/s, starts at (0,0).– C) Plot y(t) vs t

time y(t)

0 0m

1 12.5m

2 20m

3 22.5m

4 20m

5 12.5

10 -100

22 5.2152

1)( ttattvty oy


Example ProblemD) Plot object trajectory

• Choose coordinate system

time x(t) y(t)

0 0m 0m

1 10m 12.5m

2 20m 20m

3 30m 22.5m

4 40m 20m

5 50 12.5

10 100 -100

25.215)( ttty

ttx 10)(

2025.5.1)( xxxy


Relative Velocity

• So far we have looked at adding displacement vectors

• May also find situations where we need to add velocity vectors


Relative Velocity

• You might remember

5m/s

25m/s


Relative Velocity

• In this case, our hero would presumably prefer not to be decapitated by the bridge

• So we are interested in his velocity relative to the bridge• He is on a train moving at +25 m/s relative to the bridge• His velocity relative to the train is -5m/s• So his velocity relative to the bridge is:


Relative Velocity (Example 1)

• Imagine you are on a barge floating down the river with the current

• You walk diagonally across the barge with a velocity

• What is your velocity with respect to the water?

• With respect to the river bank?

iv sm

river

3

jiv sm

sm ˆ2ˆ2bargeon walk


Another River Problem

• A boat’s speed in still water is 1.85m/s. If you want to directly cross a stream with a current 1.2m/s, what upstream angle should you take?

4.40


In Class Demo

• I am going to shoot the rocket into the classroom standing still. – Where will it land with respect to my position?

• A) In front of me?• B) Behind me?• C) Same position?

x

y

?


In Class Demo

• I am going to shoot the rocket into the classroom walking forward. – Where will it land with respect to my position?

• A) In front of me?• B) Behind me?• C) Same position?

x

y

vo

?


Today We Learned….

• Vector kinematics– Displacement vector – Average velocity vector– Inst. Velocity vector– Average acceleration vector– Inst. Acceleration vector– Vector equations of motion

• Relative Velocity

department of physics and applied physics 95.141, f2009, lecture 4 physics i 95.141 lecture 4...

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average velocity vector

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