Two atomic clocks are synchronized. One is put aboard a...

Two atomic clocks are synchronized. One is put aboard a spaceship that leaves Earth at $t=0$ at a speed of $0.750 \mathrm{c}$. (a) When the spaceship has been traveling for $48.0 \mathrm{h}$ (according to the atomic clock on board), it sends a radio signal back to Earth. When would the signal be received on Earth, according to the atomic clock on Earth? (b) When the Earth clock says that the spaceship has been gone for $48.0 \mathrm{h}$, it sends a radio signal to the spaceship. At what time (according to the spaceship"s clock) does the spaceship receive the signal?

Key Concepts

Inertial Frames and Relativity of Simultaneity

Inertial frames are reference frames in which objects move at constant velocity, and special relativity shows that events which are simultaneous in one inertial frame may not be simultaneous in another due to the relativity of simultaneity. This concept is essential for analyzing how different observers, moving relative to one another, measure time and order events, highlighting the interplay between relative motion, synchronization, and the propagation of signals.

Time Dilation

Time dilation is a key concept in special relativity which states that time intervals measured in a moving system (relative to an observer) will appear longer than those measured in the observer’s own rest frame. This effect becomes very pronounced at speeds approaching the speed of light and is essential for understanding differences in elapsed time as recorded by clocks in relative motion.

Lorentz Factor

The Lorentz factor, denoted by ?, quantifies the amount of time dilation (and length contraction) in special relativity. It is defined by the equation ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light. This factor is critical for calculating relativistic effects on time and space and is a central tool in solving problems involving high-speed travel.

Signal Propagation Delay

Signal propagation delay refers to the finite time it takes for light or any electromagnetic signal to travel from one point to another. In relativistic scenarios, even if events occur at the same time in one frame, the actual observation of these events can be delayed because signals travel at a finite speed, which must be accounted for when comparing time measurements between different observers.

Transcript

00:01 Okay, so for question a, we can come up with such equation with delta t is equal to gamma times delta t0.

00:10 And gamma is the lawrence factor, which is equal to 1 over square root, 1 minus v squared over c squared.

00:18 And we know delta t0, which is the time that was measured on a spaceship, is 48 hours.

00:26 And we know the speed of spaceship is 0 .75c.

00:32 Therefore, we can have delta t, i'm sorry, delta t is equal to, 1 over square root 1 minus b square over c square times delta t 0 which is equal to 48 hours divided by square root 1 minus 0 .75c square and over c square and this will give us 48 hours divided by square root 1 minus 0 .75c square and 0 .75 and n square, which is about $72 .6.

01:23 Okay? so this is the time on earth.

01:30 And remember, in order for the signal to catch out the spaceship, we need to cover the distance that was already traveled by the spaceship.

01:41 And so during this time, it also caused some times to cover this distance.

01:46 We know the distance that was traveled by the spaceship is, 72 .6 hour, which is the time, and then times the speed of spaceship, which is 0 .75c.

01:59 Okay, let's say this distance is equal to d.

02:02 Okay? when we know the time, the cost to travel this much distance is t equal to 72 .6 hours times 0 .75c.

02:21 So the time for the signal to catch up to cover this distance is this much distance here, divide by c.

02:27 Because the speed of signal is speed of light, which is the sea.

02:31 And this will give us 54 .5 hours.

02:38 Okay, so we take 54 .5 hours for the signal to cover up the distance that was already traveled by the spaceship.

02:46 Therefore, the total time, t total, is equal to the time on earth, which is 72 .6 hour plus 54 .5 hours, okay? which is equal to 127 .1 hours.

03:18 So it takes 127 .1 hours to reach the spaceship according to the atomic clock on earth.

03:27 And for question b, you say that according to the atomic clock on earth, 48 hours passed after the spaceship departed from the earth.

03:38 So we have, in this case, let's say, delta t is equal to, 48 .0 hours.

03:49 Okay.

03:50 And after 48 .0 hours, the people on earth send a signal to a spaceship, which means that it takes, let's say, delta t2, and this is the time for this, for the signal to catch up a spaceship according to earth's atomic clock, okay? which is the total distance divide by the speed of signal.

04:23 And we know that since 48 hours was passed already, that means that the spaceship already traveled for like 48 hours, so there will be some distance traveled by the spaceship, okay? which we have the total distance is still d for the spaceship divided by the speed of spaceship, which is 0 .75c.

04:50 And let's say for this time is equal to delta t3.

04:56 Therefore we can have the relationship between d2 and d2 which is d2t3 is equal to d2 plus 48 .0 .0 .40 hours.

05:15 Okay...

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