define proper time interval as that a moving clock appears to go slow to observer
Answers
TIME DILATION
Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.
PROPER TIME
Proper time Δt0 is the time measured by an observer at rest relative to the event being observed.
For EXAMPLE CALCULATING
Δ
t
FOR A RELATIVISTIC EVENT:
Q HOW LONG DOES A SPEEDY MUON LIVE?
Suppose a cosmic ray colliding with a nucleus in the Earth’s upper atmosphere produces a muon that has a velocity v = 0.950c. The muon then travels at constant velocity and lives 1.52 μs as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal clock.) How long does the muon live as measured by an Earth-bound observe ?
Figure. A muon in the Earth’s atmosphere lives longer as measured by an Earth-bound observer than measured by the muon’s internal clock.
Strategy
A clock moving with the system being measured observes the proper time, so the time we are given is Δt0 = 1.52 μs. The Earth-bound observer measures Δt as given by the equation Δt = γΔt0. Since we know the velocity, the calculation is straightforward.
Solution
Identify the knowns: v = 0.950c, Δt0 = 1.52 μs
Identify the unknown: Δt
Choose the appropriate equation. Use Δt = γΔt0, where
γ
=
1
√
1
−
v
2
c
2
.
Plug the knowns into the equation.
First find γ.
γ
=
1
√
1
−
v
2
c
2
=
1
√
1
−
(
0.950
c
)
2
c
2
=
1
√
1
−
(
0.950
)
2
=
3.20
Use the calculated value of γ to determine Δt.
Δ
t
=
γ
Δ
t
0
=
(
3.20
)
(
1.52
μ
s
)
=
4.87
μ
s
Discussion
One implication of this example is that since γ = 3.20 at 95.0% of the speed of light (v = 0.950c), the relativistic effects are significant. The two time intervals differ by this factor of 3.20, where classically they would be the same. Something moving at 0.950c is said to be highly relativistic.
Hope this is help you...
