Environmental Engineering Reference
In-Depth Information
S
′
S
y
y
′
U
u
q
x
x
′
Figure 6.12 A particle moves in
S
at an angle
θ
to the
x
-axis at speed
v
.
Example 6.3.2
A particle moves with speed v at an angle θ to the positive x-axis
in the frame S. What is the direction of the particle in the frame S
given that S and
S
move with relative speed U along their common x direction?
Solution 6.3.2
Figure 6.12 illustrates what is going on. Using the velocity addition
formulae we can write down the components of the velocity of the particle in S
:
v
x
+
U
v
x
=
Uv
x
/c
2
,
1
+
v
y
v
y
=
Uv
x
/c
2
)
.
γ(U)
·
(
1
+
Using
tan
θ
=
v
y
/v
x
, v
x
=
v
cos
θ and v
y
=
v
sin
θ gives
U
2
/c
2
)
1
/
2
v
sin
θ
U
(
1
−
tan
θ
=
.
+
v
cos
θ
c then
tan
θ
→
This is an interesting result: if U
→
0
regardless of θ . This effect
would only happen in classical theory as U
→∞
.
PROBLEMS 6
6.1 A spaceship moves relative to the Earth at a speed of 0
.
93
c
. If a person on
Earth spends 30 minutes reading the newspaper, how long have they been
reading according to someone on the spaceship?
6.2 Pions are elementary particles, which decay with a half-life of 1
.
8
10
−
8
sas
measured in a frame in which the pions are at rest. In a laboratory experiment,
a beam of pions has a speed of 0
.
95
c
. According to an observer in the lab,
how long does it take for half of the pions to decay? Through what distance
will they travel in that time?
6.3 An alien spacecraft is flying overhead at a great distance. You see its search-
light blink on for 0.190 s. Meanwhile, on board the spacecraft, the pilot
observes that the searchlight was on for 12.0 ms. What is the speed of the
spacecraft relative to the Earth?
6.4 In the previous question, why was it necessary to state that the spacecraft
was at a great distance overhead? Suppose that the same spacecraft is flying
at 0
.
998
c
but this time at ground level and directly away from you. If the
pilot once again turns the searchlight on for 12.0 ms, how long now does the
searchlight appear to stay on according to your watch?
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