A muon is created 55.0 km above the surface of the earth (as...

A muon is created 55.0 km above the surface of the earth (as measured in the earth's frame). The average lifetime of a muon, measured in its own rest frame, is 2.20 $\mu$s, and the muon we are considering has this lifetime. In the frame of the muon, the earth is moving toward the muon with a speed of 0.9860$c$. (a) In the muon's frame, what is its initial height above the surface of the earth? (b) In the muon's frame, how much closer does the earth get during the lifetime of the muon? What fraction is this of the muon's original height, as measured in the muon's frame? (c) In the earth's frame, what is the lifetime of the muon? In the earth's frame, how far does the muon travel during its lifetime? What fraction is this of the muon's original height in the earth's frame?

Key Concepts

Special Relativity

Special relativity is the framework for understanding how space and time are interconnected when objects move at speeds close to the speed of light. It introduces the concept that measurements of time and distance are relative to the observer’s inertial frame and must be transformed when switching between these frames.

Proper Time

Proper time is the interval of time measured in the rest frame of an object, where the object is observed at rest. It is the shortest time between two events and serves as a reference for understanding time dilation when comparing it to the time intervals measured in other moving frames.

Time Dilation

Time dilation is the phenomenon wherein a clock moving relative to an observer ticks slower compared to a clock at rest in the observer's frame. This effect, which becomes significant at speeds approaching the speed of light, explains differences in measured lifetimes between frames in relative motion.

Length Contraction

Length contraction is the effect in which an object's length parallel to the direction of relative motion is measured to be shorter by an observer in a different inertial frame than measured in the object’s rest frame. It is a key concept in understanding how distances are perceived differently between moving frames.

Lorentz Factor

The Lorentz factor, often denoted by gamma, quantifies the amount of time dilation and length contraction experienced due to an object's motion at relativistic speeds. It is defined as 1 divided by the square root of (1 minus the square of the velocity over the speed of light) and is fundamental to all calculations in special relativity involving transformations between frames.

Transcript

00:01 This is chapter 37 problem number 48.

00:03 A muon is created 55 kilometers above the surface, a measure from the earth.

00:09 So our proper length is 55 kilometers.

00:14 Average lifetime over muon is also given to us.

00:17 Average lifetime over muon in its own frame is given to us.

00:22 So for the muon, it's going to be the proper time, right? 2 .2 microseconds.

00:28 So in the frame of the meon, earth is moving towards the muleon with the speed of .986c.

00:41 Now in part a, we are asked to calculate in mewon's frame.

00:48 What is its initial height? what is the initial height? remember, initial height of the mion is given to us as 55 kilometers in the earth's frame.

01:00 And now we are to calculate it in the mion's frame.

01:03 So the equation that we're going to use is l equals l0 over gamma, right? in this case, we know what l0 is.

01:13 That's the proper length.

01:15 It's 55 kilometers, so let's convert kilometers into meters times 1 over gamma factor.

01:24 1 over gamma factor is square root of 1 minus v squared, right? v is 0 .986c square over c squared.

01:38 So when we do with the algebra, what we find is 9 .17 times center of 3 meters.

01:44 So again, let me explain you the gamma factor here.

01:48 The gamma factor is 1 over square of 1 minus v squared over c squared.

01:51 That's the formula for it.

01:53 Since we have 1 over gamma, then if we take the inverse of it, that equals to 1 over v square over c squared.

01:59 And that's where this term comes from that you see here.

02:02 So in the mion's frame, the initial height is 9 .17 times 10 to 0 .3 meters...

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