A plane trip lasts 8.0 h ; the average speed of the plane during the flight relative to Earth i

A plane trip lasts 8.0 h ; the average speed of the plane...

Key Concepts

Lorentz Factor

The Lorentz factor, defined as gamma = 1/sqrt(1 - (v/c)²), quantifies the degree of time dilation and length contraction experienced by an object moving at velocity v relative to the speed of light c. It is central to calculating the differences in elapsed time between different inertial frames in relativistic scenarios.

Time Dilation

Time dilation is a phenomenon predicted by special relativity where time as measured in a moving frame appears to pass more slowly compared to a stationary observer’s frame. This effect becomes significant at speeds approaching the speed of light and is calculated using the Lorentz factor.

Special Relativity

Special relativity is a theory in physics that addresses the behavior of objects moving at constant high speeds relative to each other. It introduces the idea that time and space are not absolute, but rather relative to the observer’s frame of reference, leading to phenomena such as time dilation and length contraction.

Transcript

00:01 In this problem, we have a plane making a trip of eight hours to a destination at a speed of 220 meters per second.

00:09 Now, this picture here is in the perspective of the earth frame.

00:14 Notice the earth frame does not have an arrow on it, but the plane does.

00:19 It's from the perspective of an earth, the earth frame.

00:23 So think of an observer at the origin here, watching all this.

00:28 The destination is also part of the earth frame.

00:32 It's at rest in the earth frame.

00:34 Now, let's look at this from the standpoint of the perspective of a passenger on the plane.

00:42 Let's go back.

00:44 Here is at the original airport.

00:48 The two synchronized their clocks, t equals zero on both their clocks, and they're together.

00:53 Now, from the perspective of the passenger on the plane, again, think of that plane, that passenger at the origin of his frame.

01:01 So here's going to be the plane frame.

01:06 Here's going to be the earth frame and here's the destination.

01:20 Now, from the standpoint of that passenger on the plane, from this, the plane frame, he was originally relative to the earth, the plane was moving to the right with speed v.

01:30 He will say the earth is moving to the left.

01:32 Remember, this is a new perspective, it's not new physics.

01:36 It's just a matter how they're describing it.

01:38 The earth will describe it one way, the plane will describe it another.

01:41 The plane will say the earth is moving to the left or the earth will say the plane is moving to the right.

01:45 So the earth to the plane is moving to the left with speed b.

01:49 That also means the destination is seen by the plane to be moving to the left, the speed v.

01:55 It's fixed to the earth.

01:56 It's at rest in the earth frame.

02:00 So, again, from the notice, the frame perspective is on, has no arrow on it.

02:06 He thinks everybody else is in motion.

02:09 He's at rest.

02:10 Everybody else is doing the motion to make this all work out the way it's supposed to, the way the physics is saying it is.

02:15 All the same.

02:18 Same physical situation, just a matter of the perspective.

02:23 Now, in these problems, the relative problems, it's important to be able to distinguish to tell which frame gives you proper time, dilated time, proper length, contracted length.

02:35 Now, in this problem, we only worry about proper time and dilated time.

02:39 Now, let's understand the definition of proper time.

02:42 The two events you are timing must take place at the same point so you can use a single time, single clock to measure their time.

02:56 So let's look at this.

02:59 The plane, he had that first airport, where he left from.

03:08 He's at the, remember at the origin.

03:11 He experienced that airport at his origin.

03:15 Sometimes later, the destination will come to him again at his origin.

03:21 He's not going to, to him, he's not going to the destination.

03:24 The destination's coming to him.

03:26 That's what these arrows mean.

03:29 Totally realistic, totally true.