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
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