Physics Velocity transformation msc 2nd year

Velocity transformation

1. Velocity
Transformation
We know the components of
velocity a particle in S iend want to
find the same in S' for the same
particle.
,..,."

2. Notations
V Relative velocity between frames.
Constant as a function of time.
ii Instantaneous velocity of particle
is S. Need not be constant.
il Instantaneous velocity of particle
r: is S'. Need not be constant.

3. s

4. Special Theorv ol Belalivttv
Events Related to
Displacement
Imagine that a particle is moving
in x--direction in a frame S.
E 1: Particle found at x
1 at t
1
•
E2: Particle found at x
2 at t
2
•
~

5. Even if the velocity of particle! is
not constant
aX x
2
-x
1
~t ; t2 -~I
in the 1:imit flt tending to zero
would give the instantaneous
'Jelocity of particle in S.
:)

6. If the motion is in three­
dimension, in generat
Ll X
u ... = Lt
_ r-.o Ll t
u = Lt Ll y
,, ~ r .a Ll t
u = Lt Ll z
(◄ : ..1 r -o Ll t

7. Similarly looking at the same
particle in S ', we can define:
Ll x
1
u' -Lt
~ - JI t'
1 r~o u
u· = Lt Ll y'
Y l1 r-.o Ll t•
u: = Lt Ll z'
• .1 r·-.o Ll t'

8. Lorentz Transformation in
differential form .
.... \X' = r(&lX -v~t)
.~y· = .~y
._\Z' = ll'
~t• = r ~t -V dX
c2

9. I
X ~ X-Vt.
I I
I
X
2
: f (Xl-Vl:,
1
I I
)(2,_ }(I : 1 { Jt, .. >r, • \' ( irtJ
I
61' :: 1 [6>c. -~At
\..

10. Lorentz Transformation in
differential form.
AX'= y(.\X-V.. )
t\y' = t\y
~·=~
!.\t' = r l_\t -V -~
._ c·

11. \x
':\X' ~t -V
- -V -~t
- -·--------
~t· ~t _ V L\X l _ V dX
c
2
c
2
L
, U)c. -V
U:.i. = ---
l-vu,,,
c2

12. .\y
.~y' .\y ~t
= -----= , _______ _
~t·
0
1-V ~
r c2 _
u' = __ u ..... r __
V
1-vu,
r c2

13. Soec1ali Theorv ol RelaUvltJ
Velocity Transformation
Equations
, _ U,.-V
U~. ----
u' =
V
::)
l-vu.,.,
Cl
u,,
r 1-
VU,
c2
. u'
,· =
z
r
~-.

l
ti, ..
vu!t.
1-
c2
.I
.,..
:1
j

14. Inverse Velocity
Transformation
U' + V
u = ----· -
~- vu'
1 + 11
c2
u'
y
U =,----·u , • =
y vu' ·
u:
,
1
~
y + 2
C
1
VUK
y + ')
c·
)

15. Comment
One c.an show that
• If u<c in S, u<c in S' also
irrespective of v.
• If u=c in 5, u=c in S' also
' irrespective of v.