A spaceship resting on Earth has a length of 35.2 m. As it departs on a trip to another planet,

A spaceship resting on Earth has a length of 35.2 m. As it...

Key Concepts

Special Relativity

Special relativity is the theory that describes the physics of objects moving at speeds close to the speed of light, introducing effects such as length contraction and time dilation. It fundamentally changes our understanding of space and time, showing that measurements of lengths and time intervals depend on the relative motion between the observer and the object.

Length Contraction

Length contraction is the phenomenon in which the length of an object moving relative to an observer is measured to be shorter than its proper length (the length measured in the object’s rest frame). This contraction only occurs along the direction of motion and is described by the Lorentz factor.

Lorentz Factor

The Lorentz factor, often denoted by gamma (?), is a key quantity in special relativity that quantifies the amount of time dilation and length contraction experienced by a moving object. It is defined as ? = 1/sqrt(1 - v²/c²), where v is the relative velocity and c is the speed of light.

Time Dilation

Time dilation is the effect whereby a clock moving relative to an observer is observed to tick slower than a clock in the observer’s own rest frame. The time interval measured by the moving clock (proper time) is shorter than the time interval measured by the stationary observer by a factor of the Lorentz factor.

Transcript

00:01 In this problem, we are given seemingly disjoint information about links of the spaceships, about how long an astronaut exercise is, and doesn't seem to go together directly, but it does.

00:16 It has one common link, and that is v, which links to gamma, which connects all the, which is seen in all the about the artistic formula.

00:25 So that's, that's the common goal because they did not give us v.

00:30 They gave us links.

00:31 So let's talk about, this now.

00:34 At the initial time when the ship is at rest on the earth, so that means the earth and the spaceship frame are one of the same, effectively, at this point.

00:44 And the earth -based observer says it's got length to 35 .2 meters.

00:50 That also then would be somebody on the ship, would say 35 .2 meters.

00:55 And seeing the ship is at rest, either frame you want to talk about, this is the proper length.

01:00 That's the proper length.

01:01 I put a zero on the le -0.

01:03 You'll see over here, i got an ld again to tell us that that was special measurement at that initial time.

01:12 Nothing was in motion yet.

01:16 So that's the proper length.

01:18 Now we come over here.

01:19 It's because they were in the both frames.

01:21 It's the same.

01:23 One really frame, really.

01:25 And then we come over here, and now the ship's in motion with a speed beat.

01:29 And we are told that an earth -based observer now says it's got a length 30 .5 meters.

01:34 So this earth -based observer is seeing a moving ship.

01:38 So this is actually the contracted length.

01:48 And the spaceship frame, it gives you still, as always, the proper length, because the ship is at rest in that frame.

01:59 So this is l.

02:02 This is the contracted length now.

02:06 Now, let's use that to get gamma.

02:09 Our formula for length contraction, l equals l -0 over gamma, which gives me gamma is equal to l -e.

02:17 I mean l -s, i should say, l -s over l -e.

02:24 Now, could we calculate that number, certainly, but it just makes more work for you.

02:32 It's more error -prone, does not give, and also, maybe most important, it gives you no meaning.

02:39 As you work through these problems, you don't see how things connect when all you're doing is just punching and running down numbers.

02:48 So now we have gamma.

02:50 We could, like i said, we could calculate, we know l -s, we know l -e, so we have it.

02:56 And now we get to the point about the astronaut exercise.

03:02 And so we're going to have two times to worry about.

03:04 The time that the astronaut says and the time that the earth -based observer says.

03:10 And this one is given to us 22 .2 minutes.

03:17 Now, which one's the proper time and which one is the proper length? and ignore any, let's just assume that the astronaut is doing sit -ups at one location...

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