Special Relativity - Twins sending light pulses

[jinksys]

Homework Statement

Twin A is on earth. Twin B is aboard a spacecraft that leaves Earth toward a start 12LY away at 0.6c.
Both twins send light pulses every year according to their clocks.
How many pulses does A send and receive and how many does send and receive if B makes a round trip to the star and back?

Homework Equations

The Attempt at a Solution

For twin A I need to know how long it takes B to complete the round trip.

time = d/r = 24LY/0.6c = 40 years

Twin a would send 39 pulses since B would arrive at year 40.

For twin B, the universe is moving past him at 0.6c so I need to compute the length of the trip according to his point of reference.

L = $$L_{0}$$$$\sqrt{1 - 0.6^2}$$ = 9.6LY

t = d/r = 18.6LY/0.6c = 32 years

Twin b would send 31 pulses since b would arrive at year 32.

 

[mark726]

However, according to the concept of time dilation in special relativity, time moves slower for objects in motion. This means that twin B would experience time at a slower rate compared to twin A, and therefore would not send or receive the same number of pulses as twin A.

To calculate the exact number of pulses sent and received by twin B, we can use the time dilation equation:

t' = t/\sqrt{1 - v^2/c^2}

Where t' is the time experienced by the moving object (twin B), t is the time experienced by the stationary object (twin A), v is the velocity of the moving object (0.6c in this case), and c is the speed of light.

For twin B's round trip, we can calculate the time experienced by twin B as:

t' = 40 years/\sqrt{1 - (0.6c)^2/c^2} = 66.7 years

This means that according to twin B's clock, the round trip would take 66.7 years. Therefore, twin B would send and receive 65 pulses (66.7 - 1 for the initial pulse) during the round trip.

In summary, the total number of pulses sent and received by twin A would be 39, while the total number of pulses sent and received by twin B would be 65. This is due to the difference in the experience of time between the two twins, as predicted by the theory of special relativity.