Question

A starship takes 3.0 days to travel between two distant space stations according to its own clocks. Instruments on one of the space stations indicate that the trip took 4.0 days. How fast did the starship travel, relative to that space station?

Short answer

Answer: The relative speed between the starship and the space station was approximately 1.29 * 10^8 m/s.

Step-by-step solution

Identify the given information

We are given: 1. Time experienced by the starship: \(t_{starship} = 3.0 \, days\) 2. Time experienced by the space station: \(t_{station} = 4.0 \, days\) Our goal is to find the relative speed, \(v\), between the starship and the space station.

Convert time to seconds

First, we need to convert both given times to seconds, as follows: 1 day = 24 hours = 1440 minutes = 86400 seconds \(t_{starship} = 3.0 \, days * 86400 \, s/day = 259200 \, s\) \(t_{station} = 4.0 \, days * 86400 \, s/day = 345600 \, s\)

Apply the time dilation formula

According to the time dilation formula in special relativity, we have: \(t_{station} = \frac{t_{starship}}{\sqrt{1 - \frac{v^2}{c^2}}}\) where \(t_{station}\) is the time experienced by the space station, \(t_{starship}\) is the time experienced by the starship, \(v\) is the relative speed between the starship and the space station, and \(c\) is the speed of light (approximately \(3.0 * 10^8 \, m/s\)). We need to solve this equation for the relative speed \(v\).

Rearrange the time dilation formula

First, divide both sides of the equation by \(t_{starship}\): \(\frac{t_{station}}{t_{starship}} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) Square both sides: \(\left(\frac{t_{station}}{t_{starship}}\right)^2 = \frac{1}{1 - \frac{v^2}{c^2}}\) Next, isolate the term containing the relative speed \(v\): \(\frac{v^2}{c^2} = 1 - \frac{1}{\left(\frac{t_{station}}{t_{starship}}\right)^2}\) Finally, solve for the relative speed \(v\): \(v = c * \sqrt{1 - \frac{1}{\left(\frac{t_{station}}{t_{starship}}\right)^2}}\)

Calculate the relative speed

Now, plug in the values of \(t_{starship}\), \(t_{station}\), and \(c\) to find the relative speed \(v\): \(v = (3.0 * 10^8 \, m/s) * \sqrt{1 - \frac{1}{\left(\frac{345600 \, s}{259200 \, s}\right)^2}}\) \(v \approx 1.29 * 10^8 \, m/s\) Therefore, the starship traveled at a relative speed of approximately \(1.29 * 10^8 \, m/s\) with respect to the space station.