9/2/2009USF Physics 100 Lecture 31 Physics 100 Fall 2009 Lecture 3 Kinematics in One Dimension Gravity and Freely Falling Bodies

9/2/2009 USF Physics 100 Lecture 3 1

Physics 100

Fall 2009

Lecture 3

Kinematics in One Dimension

Gravity and Freely Falling Bodies

http://terryspeaks.wiki.usfca.edu


Thanks for the cartoon to Moose’s, 1652 Stockton St., San Francisco, CA


Agenda (1) Administrative Prof. Benton is in Germany Substitute Lecturer this week: Terrence A. Mulera – HR 102 Office Hours: Today 1-2 PM and by appointment Contact Information:

e-mail: [email protected] Phone: (415) 422-5701

1st Lab week of 14 September Homework? Syllabus is on wiki


Agenda (2) Today’s LectureKinematics in One Dimension

•Define Kinematics•Displacement•Velocity and/or Speed•Acceleration•Gravity and Falling Bodies

Tools for More Than One Dimension•Trigonometry Review, Vectors•Dave’s Short Trig Course, http://www.clarku.edu/~djoyce/trig/

Movie


Kinematics

Kinematics Description of motion with no reference to forces

Dynamics Effect of forces on motion

Kinematics + Dynamics → Mechanics


Displacement

Vector pointing from an object’s initial position to its final position

In 1-D, say x

x0 x

x

Magnitude |x|, is shortest distance from x0 to x or vice-versa. Scalar

Only thing vectorial in 1-D is ±. Also true for velocity and acceleration in 1-D.

Units: Length, e.g. m, cm, km, ft, in, miles, etc.


VelocityTime rate of change of displacement

x

vt

Vector

Note that velocity is an instantaneous quantity and can itself vary with time

d x tv t

dt

(Calculus fans only)

Digression on derivatives and tangent lines


Average velocity x

vt

The magnitude of velocity is called speed. It is a scalar which reveals nothing about the direction of motion

Typical units of both velocity and speed are m/sec, ft/sec, etc.

total distance traveled s

t

length

time


AccelerationTime rate of change of velocity

v

at

Vector

Note that acceleration is an instantaneous quantity and can itself vary with time

2

2

d v t d x ta t

dt dt

(Calculus fans only)

Time rate of change of acceleration is called jerk Time rate of change of jerk is called surge


Constant Acceleration

t

vf

v0

slope = a 0 0 v t v v v a t v

0 0 + x t x v t v t

For constant acceleration, v above is really the average value < v>

tot

2

vv

20 0

1 +

2x t x v t a t


20 0

1 +

2x t x v t a t


Summary of Units

Displacement length m

Velocity length / time m / secAcceleration length / time2 m / sec2

Quantity Type of Unit Example

Dimensional Analysis:Units in an equation expressing a physical process must “make sense”.

m (m / sec) secx vt e.g makes sense

2(m / sec)

m m / secsec

vx

t makes no sense


20 0

1 +

2x t x v t a t

2 2m m + m / sec sec (m / sec )sec m


Gravitational Acceleration1 2

2ˆ

m mF G r

r

We are getting a bit ahead of ourselves here talking about forces

Near the surface of the Earth1

2ˆ E

E

m mF G r

r

For falling distances small w.r.t. the Earth’s radius, rE

2ˆ const. units of accelerationE

E

mG r

r

Acceleration due to gravity, g

(-) downward


g 9.8 m /sec2

980 cm / sec2

32 ft /sec2{ Acceleration due to gravity on Earth

Different for other celestial objects depending on their mass and size

2ˆ O

O

mG r

r

Earth 1

1Earth's moon

61

Mars 3

Jupiter 3

Our Sun 28

g

g

g

g

g

Approximate values:


Freely Falling Bodies

Acceleration due to gravity: g = - 9.8 m/sec2

Dropping an object from a height. After a time t

(constant)

fv a t

g

Units: (m/sec2) (sec) → m/sec

Velocity:

Distance dropped: y = ½ g t2

Units: (m/sec2) (sec2) → m


Throw an object straight up with velocity v0

How long does it stay up? How high does it go?

Symmetry => time going up = time coming down

initial velocity = - final velocity velocity at apex = 0

0 0 v t v v v a t

Rearranging and

Remember that g is (-). Then the time comes out (+)

0 0 time to reach apex

total time is twice this

v v vt

g g

a g


Again, remember that g is (-). Then the height comes out (+)

y v t

00

1 and at the apex

2

v vv v v t v o

g

00

1

2

v vy v v

g

2 2 20 0

2 2

v v v

g g


Trigonometry ReviewAngles:

(+) counter-clockwise

(-) clockwise

Units:

360º in a complete circle

2 radians. Arc length in terms of radius

Note that + 360º or + 2 rad. = . is periodic.


Triangles:

a

b

c

BA

C

a + b + c = 180º or radians

Right Triangle:

Pythagorean Theorem:

B2 = A2 + C2 B hypotenuse A side opposite C side adjacent w.r.t.


Trigonometric Functions:

sin = side opposite / hypotenusecos = side adjacent / hypotenuse tan = side opposite / side adjacent cot = side adjacent / side opposite sec = hypotenuse / side adjacent csc = hypotenuse / side opposite

Note: The co-functions are the functions of the complement of .

e.g. cos = sin (90º - ), etc.


sin t

cos t

Periodic

Phase difference of 90° 0r /2


Applications:

Measurement of Height:

h

dMeasure d and , then h = d tan


Vectors

Scalars: Magnitude only e. g. mass, energy, temperature

Vectors: Magnitude and direction e. g. force, momentum

Graphically:Length represents magnitudeOrientation represents direction


Graphical Addition of Vectors

A

B C

The parallelogram of forces

C A B


Component Addition of Vectors

x-axis

y-axis

Ax

Ay

A

Ax = A cos A

Ay = A sin A

Similarly, Bx = B cos B, etc.

2 2 magnitude of , x yA A A A A

C A B

A


Add the componentsCx = Ax + Bx

Cy = Ay + By

2 2 magnitude of x yC C C C

1 tan tany yC

x x

C CArc

C C


Unit Vector Notation

ˆ

ˆ

ˆ

i

j

k

Cartesian coordinates:

x

y

zcorresponding to

Unit vectors. i.e.

z

y

x

ˆˆ ˆ, ,i j kˆˆ ˆ 1i j k

ˆˆ ˆx y zA A i A j A k


Multiplication of Vectors

Scalar(a) times Vector(A):

ˆˆ ˆx y zaA aA i aA j aA k

Vector( ) times Vector( ) → Scalar(S)A

B

x x y y z zS A B A B A B A B Scalar or dot product

Note that the magnitude squared of a vector is simply

2A A A

A

B

cosA B AB

Scalar


Vector( ) times Vector( ) → Vector( )B

A

V

V A B

Vector or cross product

Note that this yields another vector (actually an axial vector)

ˆˆ ˆ

x y z

x y z

i j k

V A B A A A

B B B

x y z z y

y z x x z

z x y y x

V A B A B

V A B A B

V A B A B


C A B

sinC AB

A

C

To the plane of AB

Right Hand Rule

Direction of advance of a right hand screw


Note that the vector product is not commutative

A B B A

Again look at Right Hand Rule

C A B

‛ A

C A × B

B × A

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9/2/2009USF Physics 100 Lecture 31 Physics 100 Fall 2009 Lecture 3 Kinematics in One Dimension Gravity and Freely Falling Bodies