2 2 2 22 2s x y z c t

2 2

1

1 v c

2E mc

A Simple (?) ProblemYour instructor drops a ball starting at t0 = 11:45 from rest (compared to the ground) from a height of h = 2.0 m above the floor. When does it hit the floor?

h = 2.0 m

F ma

F mg

a g

29.8 m/s towards the

center of the Earth

g

Need to set up a coordinate system!

A poor coordinate choice

x

y

R = 6370 km

= 36.1

h = 2 m

z

Ball Starting Point:x = (R + h) cosy = (R + h) sinz = 0

Ground Starting Point:x = R cosy = R sinz = 0

vx = 0vy = 0vz = V0

vx = 0vy = 0vz = V0

V0= 30 km/s

Acceleration:ax = -g cosay = -g sinaz = 0

t = t0 = 11:45:00

Earth

Rotating coordinates

x’y

x

y’

z

z’

t = t0 = 11:45:00

Coordinate changex’ = x cos + y siny’ = y cos - x sinz’ = z

Ball Starting Point:x’ = R + hy’ = 0z’ = 0

v’x = 0v’y = 0v’z = V0

Ground Starting Point:x’ = R y’ = 0z’ = 0

v’x = 0v’y = 0v’z = V0

Acceleration:a’x = -ga’y = 0a’z = 0

Translating space coordinates

x’

y

x

y’

z z’

Coordinate change

x’ = x - Ry’ = yz’ = z

Ball Starting Point:x’ = hy’ = 0z’ = 0

v’x = 0v’y = 0v’z = V0

Ground Starting Point:x’ = 0 y’ = 0z’ = 0

v’x = 0v’y = 0v’z = V0

Acceleration:a’x = -ga’y = 0a’z = 0

R

t = t0 = 11:45:00

Time translation

x

y

z

Coordinate changet’ = t - t0

Ball Starting Point:x = hy = 0z = 0

vx = 0vy = 0vz = V0

Ground Starting Point:x = 0 y = 0z = 0

vx = 0vy = 0vz = V0

Acceleration:ax = -gay = 0az = 0

t = t0 = 11:45:00

t’ = 0

Galilean Boost

x

Coordinate changex’ = xy’ = y

z’ = z - V0t

Ball Starting Point:x’ = hy’ = 0z’ = 0

v’x = 0v’y = 0v’z = 0

Ground Starting Point:x’ = 0 y’ = 0z’ = 0

v’x = 0v’y = 0v’z = 0

Acceleration:a’x = -ga’y = 0a’z = 0

y

z

y’

z’

x’V0= 30 km/s

t = 0

Solving the problem:

x

Ball Starting Point:x = hv = 0

Groundx = 0

ax = -g

t = 0

210 0 2x x v t at 21

2h gt 0,

2ht

g

2

2 2 m

9.8 m/s 0.64 s.

Coordinate Changes: A summaryRotating Coordinates

(around z-axis)x’ = x cos + y siny’ = y cos - x sin

z’ = z

Space Translation(x-direction)

x’ = x - ay’ = yz’ = z

Time Translationt’ = t - a

Galilean Boost(x-direction)

x’ = x - vty’ = yz’ = z

Why these?

Rescaling Transformationx’ = fxy’ = fyz’ = fz x

y

x’y’

Good vs. Bad Coordinate Transforms

goodbad

The 3D distance formula

x

y 1 1 1 1, ,P x y z

2 2 2 2, ,P x y z

s

2 22s x y

2 2 221 2 1 2 1 2s x x y y z z

x 2z

y

Good vs. Bad Coordinate Transforms 2 2 22s x y z

If a coordinate transformation leaves the quantity s2 unchanged, then it must be good, and nature’s

laws are the same in the original and final systems.

Rotating Coordinates(around z-axis)

x’ = x cos + y siny’ = y cos - x sin

z’ = z

Space Translation(x-direction)

x’ = x - ay’ = yz’ = z

Rotations (any axis), Translations (any direction), and combinations of them

Distance Invariance

x

y

1 , ,P x y z

2 0,0,0P

Prove the followingThe distance between an arbitrary point P1 = (x,y,z) and the origin does not change when you perform a rotation around the z-axis

The Distance between two points does not change when you perform a rotation or a space translation

x’

y’

2P

x’ = x cos + y siny’ = y cos - x sin

z’ = z

1 , ,P x y z

2 2 22 0 0 0s x y z 2 2 2cos sin cos sinx y y x z

2 2 2 2

2 2 2 2 2

cos 2 cos sin sin

cos 2 cos sin sin

x xy y

y xy x z

2 2 2 2 2 2 2cos sin sin cosx y z 2s

Trigonometric Functions

cos

sin

sintan

cos

2 2cos sin 1

Hyperbolic Functions

12

12

cosh

sinh

sinhtanh

cosh

e e

e e

2 2

2

2

cosh sinh 1

tanhsinh

1 tanh

1cosh

1 tanh

Math Interlude

The funny thing about light . . .

Double Star

What determines the speed of light in vacuum?

Michelson Morley Experiment Laser

Detector

MirrorsThe speed of light is independent of the motion of the source

Detector

or of the observer

The funny thing about light . . .

c = 2.998 108 m/s

x

y

Galilean Boost(x-direction)

x’ = x - vty’ = yz’ = z

c’ = c + v

v

The speed of light should change as viewed by a moving observer

The speed of light is always c, independent of the motion of the source or of the observer