Relativity of Time and Length

Relativity of Time
 
Question 1: Distance traveled by the light pulse

How does the distance traveled by the light pulse on the moving light clock compare to the distance traveled by the light pulse on the stationary light clock?

Answer 1: The distance travelled by the stationary clock is longer by a factor of gamma=1.41

Question 2: Time interval required for light pulse travel, as measured on the earth
Given that the speed of the light pulse is independent of the speed of the light clock, how does the time interval for the light pulse to travel to the top mirror and back on the moving light clock compare to on the stationary light clock?

Answer 2: The time of the stationary clock is longer by (9.4 – 6.67)*10^-6 s = 2.73 *10^-6 s .

Question 3: Time interval required for light pulse travel, as measured on the light clock
Imagine yourself riding on the light clock. In your frame of reference, does the light pulse travel a larger distance when the clock is moving, and hence require a larger time interval to complete a single round trip?

Answer 3: You observe a shorter distance and a longer time interval than the stationary observer.

Question 4: The effect of velocity on time dilation
Will the difference in light pulse travel time between the earth's timers and the light clock's timers increase, decrease, or stay the same as the velocity of the light clock is decreased?

Answer 4: The difference in light pulse travel time will decrease.

Question 5: The time dilation formula
Using the time dilation formula, predict how long it will take for the light pulse to travel back and forth between mirrors, as measured by an earth-bound observer, when the light clock has a Lorentz factor (γ) of 1.2.

Answer 5: (6.67*10^-6)(1.2) = 8.004*10^-6s

Question 6: The time dilation formula, one more time
If the time interval between departure and return of the light pulse is measured to be 7.45 µs by an earth-bound observer, what is the Lorentz factor of the light clock as it moves relative to the earth?

Answer 6:

7.45*10^-6 = (6.67*10^-6) γ

γ = 7.45/6.67

γ = 1.12

Relativity of Length

Question 1: Round-trip time interval, as measured on the light clock
Imagine riding on the left end of the light clock. A pulse of light departs the left end, travels to the right end, reflects, and returns to the left end of the light clock. Does your measurement of this round-trip time interval depend on whether the light clock is moving or stationary relative to the earth?

Answer 1: Yes, the moving light clock will experience a time interval longer than the stationary clock by a Lorentz factor.

 

Question 2: Round-trip time interval, as measured on the earth
Will the round-trip time interval for the light pulse as measured on the earth be longer, shorter, or the same as the time interval measured on the light clock?

Answer 2: The round-trip time interval measured on the earth is longer by a Lorentz factor.

 

Question 3: Why does the moving light clock shrink?
You have probably noticed that the length of the moving light clock is smaller than the length of the stationary light clock. Could the round-trip time interval as measured on the earth be equal to the product of the Lorentz factor and the proper time interval if the moving light clock were the same size as the stationary light clock?

Answer 3:  Yes, since there is no length contraction the Lorentz factor is 1.

Question 4: The length contraction formula
A light clock is 1000 m long when measured at rest. How long would earth-bound observer's measure the clock to be if it had a Lorentz factor of 1.3 relative to the earth?

Answer 4: 1000m/1.3 = 769.23m


Conclusion
When objects move at speeds approaching the speed of light, length contracts and time dilates from the moving observer's perspective relative to a stationary observer's perspective.