A spaceship makes the long trip from earth to the nearest
star system, Alpha Centauri, at a speed of 0.955$c .$ The star is
about 4.37 light years from earth, as measured in earth's
frame of reference $(1$ light year is the distance light travels
in a year). (a) How many years does the trip take, according
to an observer on earth? (b) How many years does the trip
take according to a passenger on the spaceship? (c) How
many light years distant is Alpha Centauri from earth, as
measured by a passenger on the speeding spacecraft? (Note
that, in the ship's frame of reference, the passengers are at
rest, while the space between earth and Alpha Centauri goes
rushing past at 0.955$c .$ (d) Use your answer from part
(c) along with the speed of the spacecraft to calculate
another answer for part (b). Do your two answers for that
part agree? Should they?
all right. And this problem, we want to know what a trip to Alfa Centauri will look like. Two different observers. Distant self A Centauri is 4.37 light years, and the speed that the trip is going to be taken a 0.955 si. So in part they were asked how long the trip would take from an observer based on Earth. What? And this is just the usual distance divided by velocity. To get time, penis comes out to 4.58 years now in part B were asked about the length of the trip from the perspective of an astronaut on board, Um, their time is going to be moving slower relative to the observers on Earth, which means that ah, the trip appears shorter from their perspective. So, um, the 4.58 gets contracted by a factor. One over gamma. Okay, I don't want to see their cause. It was divided out. Yeah, and the sequels 1.36 years. Now, from their perspective, the distance that they cover also gets contracted. So the distance is 4.37 and it gets contracted by the same factor to do Lauren's contraction. This comes out Tio 1.3 light years. And, um, if we take this distance and divide by the speed that they're going, wait, get that It takes 1.36 years, which is the same on dancer. We got in part B, and they have to be the same for the theory to be consistent.