An unstable particle is created in the upper atmosphere from...

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist at rest on the earth's surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

An unstable particle is created in the upper atmosphere from a cosmic ray and travels straight down toward the surface of the earth with a speed of 0.99540c relative to the earth. A scientist
at rest on the earth's surface measures that the particle is created at an altitude of 45.0 km. (a) As measured by the scientist, how much time does it take the particle to travel the 45.0 km to
the surface of the earth? (b) Use the length-contraction formula to calculate the distance from where the particle is created to the surface of the earth as measured in the particle's frame. (c) In the
particle's frame, how much time does it take the particle to travel from where it is created to the surface of the earth? Calculate this time both by the time dilation formula and from the distance calculated in part (b). Do the two results agree?

Key Concepts

Inertial Reference Frames

Inertial reference frames are coordinate systems in which objects either remain at rest or continue to move at a constant velocity, absent external forces. Special relativity relies on the principle that the laws of physics, including the propagation of light and the effects of time dilation and length contraction, hold true in all inertial frames, making the comparison of physical events across different frames a central theme.

Lorentz Factor

The Lorentz factor, denoted by ?, is a key quantity in special relativity that quantifies the degree of time dilation and length contraction an object experiences when moving at a relative speed. It is defined by the equation ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light, and it is essential for transforming time and space measurements between frames.

Time Dilation

Time dilation is a fundamental concept in special relativity that describes how a moving clock, as seen by an observer in a different inertial frame, appears to tick slower than a clock at rest relative to that observer. This effect is especially significant as the relative speed approaches the speed of light, and it is quantified using the Lorentz factor, ensuring that time intervals can differ between frames in relative motion.

Length Contraction

Length contraction is the relativistic phenomenon where an object in motion relative to an observer appears shortened along the direction of motion compared to its proper length—the length measured in the object’s rest frame. This contraction is inversely related to the Lorentz factor and becomes prominent at velocities close to the speed of light, demonstrating that measurements of distance are not absolute between different inertial frames.

Transcript

00:01 This is chapter 37 problem number 12.

00:03 An unstable particle is created in the atmosphere, 45 kilometers above the surface.

00:10 We are in part a asked to calculate for a scientist on earth.

00:18 What will be the time that it took for the particle to be created and reach the surface of the earth? so in order to do that, well, let's.

00:33 Let's write down what's given to us.

00:35 First of all, 45 kilometers, the distance of where the particle is created from the surface of the earth is going to be our proper length, right? this is 45 kilometers.

00:48 And if you write it in meters, it's going to be 45 times 10 to 3 meters.

00:53 And we know the speed of the particle, 0 .9954 times the speed of light.

01:03 Now in part a, again, we were asked at least time.

01:07 So the time, as you know, it's distance over velocity.

01:12 So then the distance is l not in this case, and our velocity is given to us.

01:18 So it's going to be 45 times center curve 3 divided by 0 .9954 times c, which is 3 times 103 .8.

01:28 From here, the time that the scientist on earth is going to measure is 1 .51 times 10 to 4 4 seconds.

01:43 So that, you know, the transit time in the particles frame is the proper time, right? that's different than this t.

01:53 And part b.

01:57 So in part b, in the particles frame, we're looking at the length, the distance, right? l -length contraction equation is what we're going to use.

02:10 L equals l -0 over the gamma factor...

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