At a speed of 0.90 c, a spaceship travels to a star that is...

At a speed of $0.90 c,$ a spaceship travels to a star that is $9.0 \mathrm{ly}$ distant. a. According to a scientist on earth, how long does the trip take? b. According to a scientist on the spaceship, how long does the trip take? c. According to the scientist on the spaceship, what is the distance traveled during the trip? d. At what speed do observers on the spaceship see the star approaching them?

At a speed of $0.90 c,$ a spaceship travels to a star that is $9.0 \mathrm{ly}$ distant.
a. According to a scientist on earth, how long does the trip take?
b. According to a scientist on the spaceship, how long does the trip take?
c. According to the scientist on the spaceship, what is the distance traveled during the trip?
d. At what speed do observers on the spaceship see the star approaching them?

Key Concepts

Time Dilation

Time dilation is a phenomenon in special relativity where a clock moving relative to an observer appears to tick slower than a clock at rest with respect to that observer. This means that the elapsed time between two events will be different when measured in different inertial frames, with the moving observer recording less time passing than the stationary observer.

Length Contraction

Length contraction is a principle of special relativity which states that the length of an object moving at a relativistic speed will appear shorter along the direction of motion to a stationary observer. Conversely, the moving observer measures a smaller distance between points in the stationary frame compared to measurements made by an observer at rest relative to those points.

Lorentz Factor

The Lorentz factor, usually denoted by ? (gamma), quantifies the amount of time dilation and length contraction experienced at a given relative velocity. It is defined by the equation ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light, and it is fundamental in transforming time and spatial coordinates between different inertial frames.

Inertial Frames

Inertial frames are reference frames in which objects obey Newton's first law—moving at constant velocity unless acted upon by external forces. Special relativity relies on the concept of inertial frames, as the laws of physics, including the invariant speed of light, are the same in all inertial frames, even though time and space measurements may differ.

Transcript

00:01 Hello, everyone.

00:02 In this problem, we're given a situation where we have a spaceship that is flying towards a distant star that is measured in our frame to be nine light years away.

00:13 So the length of the trip that is going to be taken by the spaceship is measured to be nine light years.

00:20 Now, nine light years, we can estimate that distance, or we can say that that distance is the distance that light travels in nine years.

00:27 So we can actually write the distance as 9 .0 times the speed of flight, times the time interval of one year.

00:38 Then we're going to be asked to find how long this trip is going to take in our frame, how long this trip is going to take in the frame of someone on the spaceship.

00:48 What is the distance to the star according to someone on the spaceship? and then finally, what is the velocity of the star as seen? by someone on the spaceship.

01:00 So the first thing we can do is, since we're given the speed of the spaceship, we can calculate the gamma factor, the gamma factor for 0 .90c, so 90 % of the speed of flight, is equal to 2 .294.

01:16 All right, so then we can just kind of use our usual kinematic equations for calculating how long this time takes in our frame, how long this trip takes in our frame, we're given a distance in our frame and we also know the velocity of the spaceship in our frame.

01:33 So that all we have to do is divide the distance by the speed and since we've written nine light years as 9 .0 times c times one year we can write the distance as 9 .0 times c and that's going to be divided by the velocity which is 0 .90c and so that works out to be 10 years.

01:53 So in our frame this trip is going to take 10 years to complete.

01:57 Now, the question is what is this time as measured or how does this time change as measured by someone on the spaceship due to algorithmistic effects? so someone on the spaceship is someone who's actually completing this trip.

02:16 So they are kind of the inertial frame for this trip, meaning that they are the ones actually measuring their proper time for this trip.

02:23 So t prime, i .e.

02:26 The time of the trip as measured by someone in the spaceship or in the moving frame is the proper time for this event, meaning that the 10 years is the dilated time.

02:37 So in that case, we can write t as being gamma times t prime.

02:43 And so t prime is t over gamma, which means that t prime we can find just divided by gamma...

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