In the twin paradox example, Alice boards a spaceship that...

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of $0.650 c .$ This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of $0.914 c$ to establish a relative speed of $0.650 c$ with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of $0.650 c .$ This can be viewed in terms of Alice's reference frame.
a) Show that Alice must travel with a speed of $0.914 c$ to establish a relative speed of $0.650 c$ with respect to Earth when Alice is returning back to Earth.
b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

Key Concepts

Special Relativity

Special relativity is the theory established by Einstein which describes how time and space are interrelated, especially at speeds close to the speed of light. This theory introduces the idea that measurements of time and distance depend on the relative motion between observers and emphasizes that the speed of light in a vacuum is the same for all inertial observers.

Relativistic Velocity Addition

Relativistic velocity addition is the method of combining velocities in different frames of reference when dealing with speeds close to the speed of light. Unlike classical addition of velocities, this formula ensures that the resultant speed does not exceed the speed of light, accounting for the nonlinear effects predicted by special relativity.

Lorentz Factor

The Lorentz factor, often denoted as gamma (?), is a crucial component in the equations of special relativity. It quantifies how much time, length, and relativistic mass change for an object moving relative to an observer and appears in formulas like time dilation and length contraction, relating the proper time to the coordinate time.

Time Dilation

Time dilation is the phenomenon predicted by special relativity, where an observer in one inertial frame perceives the clock in another fast-moving frame to tick slower. This effect is crucial in scenarios like the twin paradox, where different observers experience different amounts of elapsed time due to their relative velocities.

Transcript

00:01 In this problem, we're dealing with alice's return trip from a space station.

00:06 But let's talk about the whole journey.

00:08 Alice leaves the earth on a spaceship traveling to the space station.

00:14 Soon as that original spaceship gets to a space station, they transport alice instantly to a ship returning to earth.

00:24 And we're going to be asked in this problem, when it talks about alice's framework reference, they're not talking about the new spaceship, we're talking about the original spaceship.

00:33 Now, this first picture is in the earth's perspective.

00:38 The original spaceship continues to move off at 0 .65c.

00:42 Alice is now moving back toward the earth.

00:48 Now, when we're going to be doing an addition of velocity formulas, we need to worry about sign.

00:55 They are not speed formulas, they're velocity formulas.

00:59 So alice is moving in the negative direction relative to the earth.

01:03 So i've got to have a negative sign in here.

01:05 So we've got to be more careful when you're doing this type of problem.

01:08 I know a lot of the problems can be set up where you just think about speeds.

01:12 That's fine.

01:13 But when you're getting more elaborate situations, then you have to be mindful that it's a one -dimensional vector.

01:20 Now, let's look at this from what really the problem is about, the perspective of the original spaceship.

01:29 Before we do that, let me remind you that the distance of space station, from the earth, that is, that space station is at rest in the earth's frame.

01:41 So that is the proper distance.

01:43 We'll need that fact in a minute.

01:46 But now let's talk about this from the standpoint of the original spaceship, which is the whole problem.

01:50 So they mean by alice's frame of reference, not her new ship, her original ship.

01:56 So the goal of this problem is to get what the velocity of alice is relative to the original spaceship on the return journey.

02:06 Now, according to the original spaceship, that means the earth is moving to the left, and so v is minus .65c.

02:15 The space station, which is at rest in the earth frame, is also moving to left, 0 .65c speed.

02:22 That's not really, the air, spatial really is out of it here, other than distance -wise.

02:27 Okay.

02:29 So, what we want, what we want is to use the formula for u, not u -pr.

02:34 We want the addition of velocity formula when it's specified for the perspective of the frame that you're saying is at rest...

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