 #1
Athenian
 143
 33
 Homework Statement:

Problem:
The planet X is far 48 lightyears from Earth. Suppose that we want to travel from Earth to planet X in a time no more than 23 years, as reckoned by clocks aboard our spaceship. At what constant speed would we have to travel? How long would the trip take as reckoned by clocks on Earth?
 Relevant Equations:

These are the equations I used. However, there are the (what feels like hundreds of) equations out there in the SR course that I find difficult knowing which one to use. In the end, I decided the below two were my best bets in solving the assignment question.
Einstein's Equation of Time Dilation:
$$\Delta t = \frac{\Delta t_0}{(1(\frac{v^2}{c^2}))^\frac{1}{2}}$$
&
Einstein's Equation of Length Contraction:
$$L = L_0 (1(\frac{v^2}{c^2}))^\frac{1}{2}$$
Homework Statement: Problem:
The planet X is far 48 lightyears from Earth. Suppose that we want to travel from Earth to planet X in a time no more than 23 years, as reckoned by clocks aboard our spaceship. At what constant speed would we have to travel? How long would the trip take as reckoned by clocks on Earth?
Homework Equations: These are the equations I used. However, there are the (what feels like hundreds of) equations out there in the SR course that I find difficult knowing which one to use. In the end, I decided the below two were my best bets in solving the assignment question.
Einstein's Equation of Time Dilation:
$$\Delta t = \frac{\Delta t_0}{(1(\frac{v^2}{c^2}))^\frac{1}{2}}$$
&
Einstein's Equation of Length Contraction:
$$L = L_0 (1(\frac{v^2}{c^2}))^\frac{1}{2}$$
To solve this, I decided to first calculate the distance between plant X and Earth (i.e. 48 lightyears) in meters.
Therefore, considering that "c" is 299,792,458 m/s, I used that number to calculate the distance light would travel in 48 years. Once calculated, I continued by calculating what the distance of 23 lightyears would be like using the given number of "c". Afterward, I plugged these numbers into Einstein's equation of length contraction to find the needed velocity for the spaceship to travel the distance of "48 lightyears" and contract it to 23 lightyears distance.
However, in the end, the velocity was  oddly  a little over 12 kilometers per second. Having a difficult time believing the number, I checked the internet and found that the Voyager 2 once traveled ~9.7 miles per second before. Thus, I am fairly certain that it is impossible for the spaceship in the question to go sailing through 48 lightyears worth of space distance in 23 years through the aid of length contraction.
In short, I feel like either I went in a completely wrong direction in solving this question or I'm "close to the mark" but just not quite getting the process in solving for the answer.
If there's anybody on the forum that could kindly assist me in understanding the process to navigate through the special relativity problem, it would be much appreciated. Of course, if this question would be more appropriate if posted on the dedicated special relativity forum here on this website, please let me know and I'll be sure to post the question there instead.
Thank you!
The planet X is far 48 lightyears from Earth. Suppose that we want to travel from Earth to planet X in a time no more than 23 years, as reckoned by clocks aboard our spaceship. At what constant speed would we have to travel? How long would the trip take as reckoned by clocks on Earth?
Homework Equations: These are the equations I used. However, there are the (what feels like hundreds of) equations out there in the SR course that I find difficult knowing which one to use. In the end, I decided the below two were my best bets in solving the assignment question.
Einstein's Equation of Time Dilation:
$$\Delta t = \frac{\Delta t_0}{(1(\frac{v^2}{c^2}))^\frac{1}{2}}$$
&
Einstein's Equation of Length Contraction:
$$L = L_0 (1(\frac{v^2}{c^2}))^\frac{1}{2}$$
To solve this, I decided to first calculate the distance between plant X and Earth (i.e. 48 lightyears) in meters.
Therefore, considering that "c" is 299,792,458 m/s, I used that number to calculate the distance light would travel in 48 years. Once calculated, I continued by calculating what the distance of 23 lightyears would be like using the given number of "c". Afterward, I plugged these numbers into Einstein's equation of length contraction to find the needed velocity for the spaceship to travel the distance of "48 lightyears" and contract it to 23 lightyears distance.
However, in the end, the velocity was  oddly  a little over 12 kilometers per second. Having a difficult time believing the number, I checked the internet and found that the Voyager 2 once traveled ~9.7 miles per second before. Thus, I am fairly certain that it is impossible for the spaceship in the question to go sailing through 48 lightyears worth of space distance in 23 years through the aid of length contraction.
In short, I feel like either I went in a completely wrong direction in solving this question or I'm "close to the mark" but just not quite getting the process in solving for the answer.
If there's anybody on the forum that could kindly assist me in understanding the process to navigate through the special relativity problem, it would be much appreciated. Of course, if this question would be more appropriate if posted on the dedicated special relativity forum here on this website, please let me know and I'll be sure to post the question there instead.
Thank you!