A cosmic ray particle travels directly over a football field, from one goal line to the other, a

A cosmic ray particle travels directly over a football...

Key Concepts

Special Relativity

Special Relativity is a theory developed by Einstein that describes the physics of objects moving at high speeds, especially speeds close to the speed of light. It establishes that measurements of time and space are relative to the observer’s frame of reference and introduces phenomena such as the relativity of simultaneity, length contraction, and time dilation.

Lorentz Factor

The Lorentz factor, commonly denoted by ? (gamma), quantifies the amount by which time, length, and relativistic mass change for an object while that object is moving. It is defined as ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light, and is crucial in transforming measurements between different inertial frames.

Length Contraction

Length contraction is a relativistic phenomenon where the length of an object moving at a significant fraction of the speed of light is measured to be shorter along the direction of motion by an observer in a different inertial frame. This effect is directly determined by the Lorentz factor and demonstrates that spatial distances are not absolute but depend on the relative motion between the observer and the object.

Time Dilation

Time dilation is a fundamental concept of Special Relativity where a clock moving relative to an observer ticks slower compared to a clock at rest with respect to that observer. This means that intervals of time, as measured in one inertial frame, may appear longer or shorter when viewed from another frame moving at high velocity relative to the first, with the relationship governed by the Lorentz factor.

Transcript

00:01 In this problem, we have a cosmic ray particle that's going to travel from one goal line to another goal line, which is normally 100 yards, which is 91 .5 meters.

00:16 Now, this is from the perspective of the earth.

00:19 The cosmic ray is moving at 0 .5c.

00:25 And first off, let's talk about the rest frame of the cosmic ray.

00:30 Which one is it? well, let's think about a car for a second.

00:36 Forget about the cosmic ray.

00:38 And you were to put on x, y, axes, stick it to the roof of that car.

00:45 So the car is at the origin of those axes.

00:48 Does the car's position ever change from x equals zero, y equal zero, relative to those set of axes? the answer is no.

00:58 That is its rest frame.

01:00 That moves with it.

01:02 So this is the rest frame of the concept.

01:10 Cosmic ray.

01:15 So the cr frame is the rest frame.

01:19 From his perspective, from this perspective, he's at rest.

01:23 Everybody else is moving.

01:24 Now, this picture, though, is not from, we'll draw that in a minute.

01:27 This is from the earth's perspective, this picture.

01:31 Now, let's talk about this length.

01:34 The two goal lines are at rest in the earth frame.

01:40 This is, so this is l .e, but more importantly, it is the proper length between the goal lines.

01:48 It is the proper length.

01:49 So the earth gives proper length.

01:56 Now, let's redraw this from the perspective of the cosmic ray frame.

02:10 Here's the cosmic ray frame.

02:12 Here's the earth frame.

02:13 Now, remember, not different physics, not different situations, a different perspective.

02:19 So, in the, when the, relative to the earth, the cosmic ray was moving to the right with speed v.

02:29 That means the cosmic ray will say the earth is moving to the left, the speed v.

02:34 Now also, the goal lines, goal lines are part of at rest in the earth frame.

02:47 So they would be moving to the left, the speed v also.

02:55 Now, let's talk about this.

02:59 This, when we think about the length between the goal lines, think about an imaginary rod.

03:06 It's got the link 91 .5 and this rod in the earth frame is at best.

03:11 Okay, so that's the proper length.

03:13 Now that same rod, if you think about it here, the same rod is now relative to the cosmic ray frame is in motion.

03:27 When you see an object in motion from your frame of reference, you will see a the contracted length.

03:34 You might say, well, isn't this obvious? you know, the other frame gave the proper length that the cr frame has to be the contracted one? true.

03:42 But let's understand where everything comes from.

03:45 He's, the cosmic ray sees that imaginary rod, which connects the two goal lines, moving.

03:54 He will see a lesser length than 91 .5.

03:57 So i'll write a here contracted length, less than the proper length.

04:05 Now, let's also, well, we don't need to, i'll talk about what's next in a second in terms of time.

04:19 The first thing it wants is this length.

04:23 And in terms of notation of your formulas, that would be not, well, not, this would just be, we already have l0...

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