Question

An astronaut is traveling in a space vehicle moving at 0.490c relative to the earth. The astronaut measures her pulse rate at 71.0 beats per minute. Signals generated by the astronaut's pulse are radioed to the earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the earth. What pulse rate does the earth-based observer measure?

Answer 1

The pulse rate as measured by an Earth-based observer of an astronaut traveling at 0.490c relative to the Earth is approximately 81.4 beats per minute, due to the effects of time dilation according to special relativity.

Step-by-step explanation:

The question involves the concept of time dilation as predicted by Einstein's theory of special relativity. Since the astronaut is moving at a velocity of 0.490c relative to Earth, and time dilation effects are observed when one frame of reference is moving at a significant fraction of the speed of light relative to another, we must apply the time dilation formula to find the pulse rate as observed by an Earth-based observer.

The time dilation formula is given by:

T' = T / √(1 - v^2/c^2)

where T' is the dilated time (pulse rate as measured by an observer on Earth), T is proper time (the astronaut's pulse rate), v is the velocity of the moving frame relative to the observer's, and c is the speed of light. In this case, the astronaut's proper pulse rate is 71.0 beats per minute. We can substitute these values into the formula to determine the pulse rate as measured by the Earth observer:

T' = T / √(1 - (0.490c)^2/c^2) = 71.0 / √(1 - 0.490^2) = 71.0 / √(0.7599) = 71.0 / 0.8717 ≈ 81.4 beats per minute

Therefore, the Earth-based observer would measure the astronaut's pulse rate as approximately 81.4 beats per minute.