Question

Astronaut mark uri is space-traveling from planet x to planet y at a speed of relative to the planets, which are at rest relative to each other. when he is precisely halfway between the planets, a distance of 1.0 light-hour from each one as measured in the planet frame, nuclear devices are detonated on both planets. the explosions are simultaneous in mark's frame. what is the difference in the time of arrival of the flashes from the explosions as observed by mark?

Answer 1

The time of arrival of the flashes from the explosions will be different for Mark due to time dilation. Mark's velocity relative to the planets will cause a time difference between the observed arrival times. The Lorentz transformation formula can be used to calculate the time dilation factor.

Step-by-step explanation:

The difference in the time of arrival of the flashes from the explosions as observed by Mark can be calculated using the concept of time dilation. According to the theory of special relativity, time is relative and depends on the observer's frame of reference. As Mark is traveling at a speed relativistic to the planets, the time measured by Mark will be different from the time measured by an observer at rest on the planets.

In this scenario, Mark is traveling halfway between the planets, so the distance to each planet from Mark is 1.0 light-hour. The explosions on both planets are simultaneous according to Mark's frame of reference. However, due to time dilation, the time of arrival of the flashes from the explosions will be different as observed by Mark.

The time dilation factor can be calculated using the Lorentz transformation formula:
t' = t * sqrt(1 - (v^2 / c^2))

Where:
t' is the time measured by Mark
t is the time measured in the planet frame
v is Mark's velocity relative to the planets
c is the speed of light

Since Mark is traveling at a speed relativistic to the planets, his velocity v will be a significant fraction of the speed of light, resulting in a noticeable time dilation effect.