A deep space probe moves away from the earth with a speed of 0.75c. An antenna on the probe requires 3.00s probe time to rotate through 1 revolution. How much time is required for 1.00 rev according to an observer on Earth?
β =0.75
t = tₒ/sqrt(1-β²) =
= 3/sqrt(1-0.75²) =4.54 s.
To determine the time required for 1.00 revolution according to an observer on Earth, we need to apply the concept of time dilation.
Time dilation is a phenomenon predicted by Einstein's theory of special relativity, which states that time appears to pass more slowly for objects moving relative to an observer. The time dilation factor is given by the equation:
time dilation factor = 1 / √(1 - (v^2 / c^2))
In this equation:
- v represents the velocity of the probe relative to the observer on Earth
- c is the speed of light in a vacuum
Given that the velocity of the probe is 0.75c, we can substitute these values into the equation and calculate the time dilation factor:
time dilation factor = 1 / √(1 - (0.75c)^2 / c^2)
= 1 / √(1 - 0.5625)
= 1 / √(0.4375)
= 1 / 0.6614
≈ 1.513
This means that time appears to be dilated by a factor of approximately 1.513 for an observer on Earth.
Now, to find the time required for 1.00 revolution according to the Earth observer, we multiply the probe time by the time dilation factor:
time required for 1.00 rev on Earth = 3.00s * 1.513
≈ 4.539s
Therefore, according to an observer on Earth, it would take approximately 4.539 seconds for 1.00 revolution of the antenna on the probe.