This document discusses concepts related to Galilean transformations and special relativity. It shows that under a Galilean transformation with velocity u: 1) The coordinates x', y', z', t' of a moving frame are related to the coordinates x, y, z, t of a stationary frame by x' = x - ut, y' = y, z' = z, t' = t. 2) Quantities like momentum and kinetic energy are unchanged under this transformation, satisfying conservation of momentum and energy. 3) The transformation satisfies the invariance of the speed of light c under the Galilean transformation.

1. @prof_Ahmad07
v
c
v≤c
v<<<c
10 -15
Fm
d
10 -10
A0
1
10-2
10-13
10-40
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2. @prof_Ahmad07
-
–
–
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3. @prof_Ahmad07
–-
s
s/
y
y/
u
x/
ut
x
o
z
o
.A
x/
/
z/
t
(x,y,z)
(x,y,z,t)
X+
o=o/
t
S/
u
t=t/
o&o/
(x/,y/,z/,t/) (x,y,z,t)
A
x / = ut – x --------------------- 1
y / = y -------------------------- 2
z / = z -------------------------- 3
t / = t --------------------------- 4
–
x = ut + x /
y=y/
z=z/
t=t/
3
1,2,3,4

4. @prof_Ahmad07
–
dt = dt /
t=t/
dx / dt / = vx / , dy / dt / = vy / , dz / dt / = vz /
dx / dt = vx , dy / dt = vy
, dz / dt = vz
then :
vx / = vx – u ---------------- 5
vy / = vy -------------------- 6
vz / = vz -------------------- 7
–
7
4
1

5. @prof_Ahmad07
s/
( m1 , m2 )
s
m1v1+m2v2 = m1v1+m2v2
m1(v1-u) + m2(v2-u) = m1(v1-u) + m2(v2-u)
m1v1 – m1u + m2v2 – m2u = m1v1 – m1u + m2v2 – m2u
m1v1 + m2v2 – u(m1+m2) = m1v1 + m2v2 – u(m1+m2)
Then : m1v1+m2v2 = m1v1+m2v2
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6. @prof_Ahmad07
1/2m1v21 + 1/2m2v22 = 1/2m1v21 + 1/2m2v22
1/2m1(v1 – u )2 + 1/2m2(v1 – u )2 = 1/2m1(v1 – u )2 + 1/2m2(v1 – u )2
1/2m1(v12 – 2v1u + u2 ) + 1/2m1(v22 – 2v2u + u2 ) = 1/2m1(v12 – 2v1u + u2 ) + 1/2m1(v22 – 2v2u + u2 )
1/2m1v12 – m1v1u + 1/2m1u2 + 1/2m2v22 – m2v2u + 1/2m2u2 = 1/2m1v12 – m1v1u + 1/2m1u2 + 1/2m2v22 – m2v2u
+ 1/2m2u2
1/2m1v12 + 1/2m2v22 + 1/2m1u2 + 1/2m2u2 – u(m1v1 + m2v2 ) = 1/2m1v12 + 1/2m2v22 + 1/2m1u2 + 1/2m2u2 –
u(m1v1 + m2v2 )
1/2m1v21 + 1/2m2v22 = 1/2m1v21 + 1/2m2v22
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7. @prof_Ahmad07
S/ S
-
S
q
S
q
/
E
F = qE
v
-
c
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8. @prof_Ahmad07
–
D
-
u
A
B
C
a
aD = aC = L
VLo = VLe + Veo
VLo = c + u
VLo
VLe
Veo
aD
aC
ty
tx
▲t
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9. @prof_Ahmad07
z/
z
x
u
/
x/
O
O
/
y
y
S/ S
S(x,y,z,t) , S/(x/,y/,z/,t/)
t = t/ = 0
O = O/
O/
O
(x/)2 + (y/)2 + (z/)2 = c2t/2
x2 + y2 + z2 = c2t2
c
(x/)2 + (y/)2 + (z/)2 - c2t/2 = x2 + y2 + z2 - c2t2
x2 + y2 + z2 = t2 ( c2 – u2 ) + 2xut ……………1
x/ = a1x + a2t ………..2
t/ = b1x + b2t………...3
2
x= ut
x/= 0
0 = a1ut + a2t
0 = t (a1u + a2 ) = 0
a2 = - a1u ……………….. 4
2
/
x = a1x - a1ut
x/ = a1( x – ut ) ………………….5
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10. @prof_Ahmad07
z/ = z y/=y
(x/)2 + (y/)2 + (z/)2 = c2t/2
3 5
{a1( x – ut ) }2 + y2 + z2 = c2 (b1x + b2t )2
{ a12 ( x2 – 2xut – u2t2 ) } + y2 + z2 = c2 (b12x2 + 2b1b2xt + b22t2)
x2 (a12 - c2 b12) + y2 + z2 = t2(c2 b22 - a12 u2) + xt(2b1b2c2 + 2ua12)
(a12 - c2 b12) = 1 ………………6
(c2 b22 - a12 u2) = c ……………7
(2b1b2c2 + 2ua12) = 0 ………...8
6,7,8
…………. 9
…………10
K
–
2,3
x/ = K(x – ut )
y/=y
z/ = z
t/ = K(t -
x)
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9,10

11. @prof_Ahmad07
–
K=1
c
u
-
x/ = (x – ut )
y/=y
z/ = z
t/ = t
u
–-
x/ = K(x + ut )
y/ =y
z/ = z
t/ = K(t +
x)
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