Problem 98 - Fundamentals of Physics

Question

An astronaut exercising on a treadmill maintains a pulse rate of 150 per minute. If he exercises for \(1.00 \mathrm{~h}\) as measured by a clock on his spaceship, with a stride length of \(1.00 \mathrm{~m} / \mathrm{s}\), while the ship travels with a speed of \(0.900 c\) relative to a ground station, what are (a) the pulse rate and (b) the distance walked as measured by someone at the ground station?

Step-by-Step Solution

Answer

The observed pulse rate is approximately 68.4 beats/minute, and the distance walked is about 8232 meters.

Step 1: Understanding Given Data

We need to find the pulse rate and distance walked as observed from the ground station. The astronaut's pulse rate is 150 beats per minute or 2.5 beats per second. The ship is traveling at a speed of 0.900c, where c is the speed of light, relative to a ground station, and the exercise time measured on the ship is 1 hour.

Step 2: Calculating Time Dilation

According to the theory of relativity, time as observed from a stationary frame (ground station) is dilated. The time dilation formula is \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( t \) is the time duration measured in the moving frame (1 hour), \( v = 0.900c \), and \( c \) is the speed of light. Calculating this gives us the dilated time \( t' \).

Step 3: Time Calculation

Using the time dilation formula, \( t' = \frac{1 \, \text{hour}}{\sqrt{1 - (0.900)^2}} = \frac{1 \, \text{hour}}{\sqrt{0.19}} \approx 2.29 \, \text{hours} \). We find that the exercise lasts approximately 2.29 hours as viewed from the ground station.

Step 4: Calculating Observed Pulse Rate

From the ground station's perspective, the time is dilated, hence the pulse rate will slow down. The pulse rate observed is \( \frac{\text{Original Pulse Rate}}{\text{Time Dilated Factor}} = \frac{2.5 \, \text{beats/s}}{\sqrt{0.19}} \approx 1.14 \, \text{beats/s} \). Converting this to beats per minute, we get approximately 68.4 beats per minute.

Step 5: Calculating Distance Walked Without Dilation

The stride length on the ship is 1 m per second, and within 1 hour, the astronaut walks \(60 \, \text{minutes} \times 60 \, \text{seconds/minute} \times 1 \, \text{m/s} = 3600 \, \text{meters}\). This is the distance covered according to the astronaut.

Step 6: Calculating Observed Distance Walked

From the ground station, the time observed is longer (2.29 hours). Hence the distance walked as seen from the ground station is: \(3600 \, \text{meters/hour} \times 2.29 \, \text{hours} \approx 8232 \, \text{meters}\).

Key Concepts

Time Dilation in Relativity

Time dilation is a fascinating concept in relativity, where time seems to stretch out or 'dilate' when observed from a different frame of reference. Consider an astronaut on a spaceship traveling at a significant fraction of the speed of light. From the astronaut's point of view, they exercise for 1 hour on a treadmill.
But for a stationary observer, like someone on Earth, this time seems longer. The key formula we use here is the time dilation formula:

• \( t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \)

Here, \( t \) is the time duration in the spaceship, \( v \) is the velocity of the spaceship (0.900c in this example), and \( c \) is the speed of light. By applying this formula, we learn that what is 1 hour on the ship actually lasts about 2.29 hours from the ground station's view. This discrepancy in time perception is a hallmark of relativistic physics, showcasing the interconnectedness of time and motion in space.

Relativity's Impact on Pulse Rate

The astronaut maintains a pulse rate of 150 beats per minute, equivalent to 2.5 beats per second while on the spaceship. However, due to the time dilation effect, this rate will appear differently to an observer on Earth. Since time appears stretched from the observer's point of view, the pulse rate effectively slows down.
The pulse rate observed from the stationary reference frame can be calculated as follows:

• To find the pulse rate observed at the ground station, divide the pulse rate by the time dilation factor: \( \frac{2.5 \, \text{beats/s}}{\sqrt{0.19}} \approx 1.14 \, \text{beats/s} \)
• Converting this to beats per minute gives around 68.4 beats per minute.

• Key point here is that as time slows down from an observer's perspective, so too does the pulse rate.

This demonstrates an important implication of relative motion: not only time but also physiological processes appear to change to an outside observer.

Distance Calculation Over Time Dilation

In this scenario, the astronaut's stride length on the treadmill is 1 meter per second. Over 1 hour, the astronaut walks 3600 meters as measured on the spaceship. But how far does this appear to an observer on Earth? Due to time dilation, the time measured from the ground station is about 2.29 hours.

To find the distance walked as observed from the ground station, we apply the concept of relative distance calculation:

• The astronaut walks 3600 meters in the spaceship's timeframe.
• As time appears longer on Earth \( (2.29 \text{ hours}) \), the distance appears longer too.
• Therefore, the observer from Earth would calculate the astronaut's distance to be approximately \( 3600 \, \text{meters/hour} \times 2.29 \, \text{hours} \approx 8232 \, \text{meters} \).

• This result shows how the perception of distance is also influenced by the relativistic effects of speed and relative movement.