An astronaut has spent a long time in the space shuttle traveling at 7.860 km / s. When he retu

step 1 In this problem, we have an astronaut and his twin. The astronaut takes off into space, travelin

An astronaut has spent a long time in the space shuttle...

Key Concepts

Lorentz Factor

The Lorentz factor, denoted by gamma (?), quantifies the degree of time dilation, length contraction, and relativistic mass increase experienced by objects moving at high speeds. It is calculated as ? = 1/?(1 - (v/c)²), and acts as a multiplier in the equations describing these relativistic effects.

Twin Paradox

The twin paradox is a thought experiment that illustrates time dilation in special relativity. It involves twins where one remains on Earth while the other travels at high speed in space; upon reunion, the traveling twin is younger than the stay-at-home twin, highlighting the relative nature of time.

Special Relativity

Special relativity is the physical theory that describes how space and time are interwoven and how they are affected by relative motion between observers. It introduces the idea that the laws of physics remain the same in all inertial frames and leads to counterintuitive phenomena such as time dilation and length contraction.

Time Dilation

Time dilation is a phenomenon predicted by special relativity where a clock moving relative to an observer ticks slower than a clock at rest in the observer's frame. This effect becomes significant at speeds approaching the speed of light and is key to understanding differences in elapsed time between moving and stationary clocks.

Transcript

00:03 In this problem, we have an astronaut and his twin.

00:07 The astronaut takes off into space, traveling basically just under eight kilometers per second.

00:13 For a certain amount of time, he comes back.

00:16 And when they compare clocks, the twin and the astronaut, compare clocks, they find that the clock that stayed with the twin on earth reads one second more than the clock on the spaceship that's to travel with the astronaut.

00:32 So that says that the twin that stayed behind is one.

00:34 Second older than the twin that left.

00:38 Now let's talk about which one is proper time, which one is dilated time.

00:43 Topper time, the two -adventure timing must take place to the same position so you can use a single clock.

00:49 The astronaut is sitting here at his origin.

00:53 Remember in his frame, his frame, he's at rest.

00:56 He experiences the spaceport where he takes off.

00:58 Whatever amount of time later he comes back to that same spaceport still at his origin.

01:05 So he experienced that's a space port.

01:06 He experienced that.

01:06 Space port at his origin, same position.

01:10 So he could use a single clock.

01:12 So he gives the astronaut frame, clock in that frame gives proper time.

01:18 The earth then gives dilated time.

01:26 Now, let me define just another quantity here just to just to make it simple algebraically, just delta.

01:34 There's no, no physical meaning.

01:36 Just i don't want to be carrying around the, the, the, the, the, the, the, the, the, um, subscripts here because it may confuse the situation in a second.

01:46 And i'll show you what i mean by that in a second.

01:48 So let's now do our time violation formula.

01:52 So we have delta t's gamma, delta t not, which in this case means delta t sub e is equal to gamma, delta t sub a, which is the same as delta t sub a over square root 1 minus v squared over c square.

02:19 We're going to need that in a second.

02:23 Now, you say, well, isn't this two equations and two unknowns? can't we just substitute for delta t .e, put it in terms of delta t .a.

02:38 Then solve for our time.

02:45 Certainly, you could do that.

02:48 One problem.

02:50 This number, look at this.

02:53 It's only eight kilometers per second.

02:55 Basically, eight times 10 to the three meters per second, compared to three times 10 to the eight.

03:00 There's a 10 to the fifth power difference between the two.

03:04 You punch in on a normal calculator, solving that equation, two equations, two unknowns, you will get basically one over zero.

03:14 Basically, it will give you no answer.

03:17 An error or infinity, you know, think about infinity.