Question

Two stationary space stations are separated by a distance of 100. light-years, as measured by someone on one of the space stations. A spaceship traveling at \(0.950 c\) relative to the space stations passes by one of them heading directly toward the other one. How long will it take to reach the other space station, as measured by someone on the spaceship? How much time will have passed for a traveler on the spaceship as it travels from one space station to the other, as measured by someone on one of the space stations? Round the answers to the nearest year.

Short answer

Answer: It takes 105 years (as measured by someone on the spaceship) and 333 years (as measured by someone on one of the space stations) to travel between the two space stations.

Step-by-step solution

Identify known values

We know that the distance between the two space stations is 100 light-years, and the spaceship is traveling at 0.950c (where c is the speed of light).

Calculate proper time that it takes for the spaceship to travel 100 light-years

To find the proper time in the spaceship's reference frame, we will use the formula $$time = \frac{distance}{speed}.$$ In our case, the distance is 100 light-years, and the speed is 0.950c. So, the proper time is $$time = \frac{100}{0.950} = 105.26 \,\text{years}.$$

Apply time dilation formula to calculate the time for someone on the space stations

The time dilation formula is given by $$t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}},$$ where \(t_0\) is the proper time in the spaceship's reference frame (105.26 years), \(v\) is the relative speed (0.950c), and \(c\) is the speed of light. Plug the values into the formula: $$t = \frac{105.26}{\sqrt{1-\frac{(0.950c)^2}{c^2}}} = \frac{105.26}{\sqrt{1-0.9025}} = 333.33 \,\text{years}.$$

Round the answers to the nearest year and answer the exercise

From our calculations, it takes the spaceship 105 years (rounded to the nearest year) to travel 100 light-years between the two space stations as measured by someone on the spaceship. However, for someone on one of the space stations, it takes 333 years (rounded to the nearest year) for the spaceship to travel the same distance due to time dilation effects in special relativity.

Key concepts

Light-Years in Physics

When we gaze at the stars, we are looking across vast distances that can be hard to comprehend. To make these distances manageable, astronomers use the term 'light-year', which is the distance light travels in one year. Light travels incredibly fast—about 299,792 kilometers per second in a vacuum.

Knowing this speed, we can calculate that one light-year equates to approximately 9.461 trillion kilometers. When we say two objects in space are 100 light-years apart, as is the case with the two space stations in our exercise, it means light would take 100 years to travel from one to the other if moving through the vacuum of space.

This measure of distance is especially useful in astronomy because it relates the concept of space to that of time, which directly ties into the implications of special relativity and understanding phenomena like time dilation.

Relativistic Speed

Entering the realm of relativistic speed means approaching or surpassing a significant fraction of the speed of light, typically denoted as 'c'. At these speeds, the normal rules of motion that we are accustomed to from classical mechanics no longer apply, which is exactly the case with our spaceship travelling at 0.950c.

Relativistic speeds bring about all sorts of intriguing phenomena – objects seem to increase in mass, lengths appear to contract, and time dilates, meaning it appears to slow down from certain perspectives. These effects become more pronounced as one gets closer to the speed of light, with our example emphasizing a major consequence of approaching these speeds: a significantly different passage of time for different observers, depending on their relative velocity.

Reference Frames in Physics

The concept of reference frames is central to understanding motion in physics. A reference frame is a perspective from which we measure and describe the movement of objects. In classical mechanics, we often consider these frames as absolutes; however, in special relativity, we must know that no single reference frame is considered special or more 'correct' than any other.

When dealing with the space stations and the moving spaceship, we're dealing with two different reference frames: the perspective of an observer at rest with respect to the space stations and the perspective from within the spaceship, which is in motion. The fundamental law of special relativity states that the laws of physics are the same in all inertial (non-accelerating) frames of reference, but the measurements of time and space will differ between frames that are in relative motion.

Lorentz Transformation

The Lorentz transformation equations are critical in bridging the gap between multiple reference frames moving at constant velocities relative to each other. Named after Hendrik Lorentz, these mathematical formulas describe how measurements of space and time by an observer in one frame relate to those by an observer in another frame.

In the context of our exercise, the Lorentz transformation allows us to calculate how time and space measurements differ between the space stations and the moving spaceship. It is through this framework that we derive the time dilation formula:
\[\begin{equation}t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}},\end{equation}\]
where \( t \) is the time experienced by the stationary observer, \( t_0 \) is the time experienced by the moving observer (proper time), \( v \) is the relative velocity, and \( c \) is the speed of light. Using this formula, we were able to determine how much time has passed for both the traveler on the spaceship and someone on one of the space stations.