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Tease Your Brain!


HansonKerman

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A vehicle moving at .927c moves in a straight line from mars to earth.

a signal moves from earth to mars.

what is the time difference in minutes between the arrivals?

Edited by Boorang
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The answer requires additional information. This includes:

1. The reference frame in which the time is measured
2. The angular alignment of the planets
3. The definition of the emission time
4. The definition of the arrival time

The Lorentz factor of an object travelling at .927c is approximately 2.666. This means that a second as measured in the object's reference frame is equal to 2.666 seconds as measured in the reference frames of Earth and Mars.

Earth's semimajor axis is approximately 1.496*10¹¹m. Mars' semimajor axis is approximately 2.2792*10¹¹m. Their distance, therefore, varies from 7.832*10¹⁰m at closest to 37.752*10¹⁰m at furthest. This is equal to 261.5–1260.5 light-seconds. An object traveling at .927c will close this distance in 282–1359.8 seconds in the reference frame of either planet (which, while not motionless, have relative motion of much smaller magnitude), but only 105.8–510.05 seconds in the object's own reference frame. Of course, the light takes less time in the object's reference frame because of apparent length contraction (light travels at c no matter the reference frame). From the object's reference frame, the light will take 105.776–472.806 seconds.

The difference in elapsed time, therefore, ranges from 21–99 seconds in the planetary reference frames and from .224–37.247 seconds in the object's reference frame.

For the question to be meaningful, we must assume that the signal and the object were emitted simultaneously. However, this has several different possible meanings. In the first, the emissions are simultaneous in some omniscient reference frame. In the second, simultaneity occurs in the Mars reference frame. In the third, simultaneity occurs in the Earth reference frame. For the first, no adjustment must be made. For the second and third, the signal or the object's light have already arrived, and so the entire travel time of the light must be added to the object or the signal, respectively.

We must also consider the differences in the measurement of the arrival. In the first case, the arrivals are measured in some omniscient reference frame. No adjustments are necessary. In the second and third, the arrivals are measured with respect to the Mars or Earth reference frames. The travel time of the light must be added to the object or the signal, respectively.

These extra cases thoroughly confuse the issue, altering the answer by as much as ±523–2521 seconds, depending on the planetary alignment. There are eighteen possible experimental arrangements, by three different variables, at any given planetary alignment.

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