(a) If you measure the ship that passes you in Problem 71 to be 24 m long, how long will the ob

(a) If you measure the ship that passes you in Problem 71 to...

Key Concepts

Relative Motion and Inertial Frames

In special relativity, measurements such as lengths depend on the observer’s state of motion. Different inertial frames (for instance, the ship’s frame and Earth’s frame) can disagree on the length of a moving object. An object measured in its own rest frame gives the proper length, while a moving observer will record a contracted length due to their relative velocity.

Proper Length

This is the length of an object measured in the frame in which the object is at rest. It represents the maximum length of the object and is not subject to relativistic contraction. For example, when observers on Earth measure their own stationary objects, they are measuring the proper length.

Length Contraction

Length contraction is a phenomenon predicted by special relativity where the length of an object moving relative to an observer is measured to be shorter than its proper length. The contracted length is given by the proper length multiplied by the factor ?(1 - v²/c²), where v is the relative speed and c is the speed of light.

Transcript

00:01 The setup for this problem, where you have two spaceships, is really given by question 71.

00:09 So it gives us the velocity of spaceship one relative to the earth, velocity of spaceship two relative to spaceship one.

00:16 You're supposed to find in that problem, the velocity of spaceship two relative to the earth, which is right here.

00:23 0 .966c.

00:25 Now, this problem takes those velocities and asks us a series of questions about links.

00:33 Seen in different frames.

00:35 So let's start in part a.

00:38 They give us that ship 1 says the length of ship 2 is 24 meters.

00:47 Now which one of these is proper length and which one is contracted length? well, ship 2 is at rest in frame s2.

00:56 This by definition then is proper length.

01:00 L2 is.

01:01 And this then is just l2.

01:02 And let's remember our form for length contraction, proper length divided by gamma is the contracted link.

01:16 Okay, so we have that information.

01:19 We don't know l2.

01:21 The end goal of part a is to find the length of that spaceship, s2, from earth -based observer.

01:30 What does the earth -based observer say? but we need to know l -nod in some manner, which is really l2.

01:41 Likewise, in this case, we know this, when we do the problem, this part where the errs and s2 are, i'm just rewriting it for that, then this is l, and this is certainly l not still.

01:53 Nothing changes.

01:54 The spaceship's rest frame is still the spaceship's s2's rest frame is still the same.

02:00 So, but let's get to, let's do the first part here.

02:04 So we have l equals l not over gamma.

02:08 So this means we're going to have l1, is it? equal to l2 over gamma, which is l2 square root of 1 minus v2 over c squared.

02:28 That's what we get between the two frames.

02:32 Now we don't know l2, so let's solve for l2, l1 over square root 1 minus v2 squared over c squared over c square.

02:45 So as we could, could we calculate there's a number? certainly, but we don't need to.

02:50 We don't need to.

02:52 We can put that into our formula here.

02:54 So now technically we know l2.

02:57 Like i said, we have all the numbers.