A cosmic-ray proton entering the atmosphere from space has a kinetic energy of 2.0 ×10^20 eV. (a

A cosmic-ray proton entering the atmosphere from space has a...

Key Concepts

Unit Conversion (Electronvolts to Joules)

This concept involves changing energy units from electronvolts (eV) to joules (J). It is crucial because energy in particle physics is often given in eV; however, many physical equations and constants (like those in mechanics) are expressed in SI units. The conversion factor is 1 eV = 1.602 × 10?¹? J, which provides a bridge between these common energy units in physics.

Gravitational Potential Energy and Work

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically defined by the equation PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. This concept is used to relate energy to the ability to do work—specifically, how much energy is required to lift an object against the force of gravity.

Special Relativity and the Lorentz Factor

In the realm of high-speed particles, special relativity becomes important. The Lorentz factor, denoted as ?, is defined by the equation ? = 1/?(1 - v²/c²) and quantifies the relativistic effects at speeds close to the speed of light (c). It is used to relate a particle's kinetic energy to its speed, ensuring that the calculations remain accurate even when velocities approach that of light.

Binomial Approximation in Relativistic Calculations

The binomial approximation is a mathematical tool used to simplify expressions that contain terms of the form (1 - x)? when x is very small. In relativistic calculations, when the Lorentz factor ? is very large and v is very near c, this approximation is applied to extract approximate values for speed without the need for extreme precision in computation, simplifying the analysis of near-light-speed motion.

Transcript

00:01 In this problem, we have a concentrated particle, which is actually a proton.

00:05 We're giving it as kinetic energy.

00:07 In the end, our goal is the speed.

00:09 We've got a few things to answer before we get to that part.

00:12 First, it wants to know what the kinetic energy is in terms of joules.

00:18 So 2 .0 times 10 to the 20 electron volts times 1 .6 times 10 and minus 19 joules over 1 electron volt.

00:30 Remember, that's the amount, this is the amount of energy charge of magnitude e would gain going through a potential difference of one.

00:44 And not important, you know, depending on the sign of the charge, it would be plus one or minus one for the potential difference.

00:51 Not important.

00:54 32 joules is the answer.

00:59 Part b wants to know if you could take this somehow and, and, you know, give it to an object, how high would it rise? so we're going to look at the kinetic energy theorem, w -toto.

01:15 We'd have the work due to this, i'll call it wp for what came from the proton.

01:22 And we still always have the work done by gravity or changing height that's always going to be there.

01:27 Now, we're going to want to go from, say, rest at a certain level.

01:34 And it's from wherever, the floor, it doesn't matter.

01:40 And up a certain distance, and also be at rest at that end distance.

01:45 So this is equal to zero.

01:46 So from rest to rest.

01:48 So you can think of the wp would be putting in 32 degrees of kinetic energy.

01:56 And during that ascent and that rising, the gravity, which opposes that upward motion, would take it all away and you'd be left back with zero.

02:06 You give it that way.

02:07 So, like i said, this is the thing.

02:10 32 joules.

02:14 So wp.

02:16 Now the work done by gravity is m minus mgh.

02:20 This is the work done by gravity, not the potential energy.

02:24 So the force is down, the displacement is up.

02:27 The cosine of the angle is cosine 180, which is minus 1.

02:32 So remember, work talks about how much you either take in, put into kinetic energy, take out kinetic energy.

02:40 Gravity is going to fight any upward motion, so it's going to be taking it out, as i was just describing.

02:46 So this is equal to zero.

02:47 So this gives me that mgh, mgh is equal to wp, implies h is equal wp over mg, 32 joules over one kilogram, 9 .8, meters per second squared.

03:20 And this works out to be 3 .3 meters.

03:25 That's how high.

03:26 You could harness that energy in some device and apply it to lifting a one kilogram object.

03:34 That's how high you could get it to go.

03:38 Part c, now we want to get to the speed.

03:40 This is not as straightforward as you may think.

03:42 You may think, oh, it's just the vs in our relativistic formulas.

03:46 We'll be able to get it.

03:47 It doesn't work out that way.

03:49 So let's start with the kinetic energy, relative is the kinetic energy.

03:53 Gamma minus 1 mc squared mc squared is the rest of energy of a proton this gives me that gamma minus 1 is equal to k over mc squared that gives me that gamma gamma is going to be equal to 1 plus k over mc squared and we can put in our numbers 1 plus 2 .0 times 10 to 20 electron volts and i'm going to give you the rest energy of the proton in electron volts.

04:37 You could have converted this to joules, use 32 for above, and use the kilogram mass of the of the proton...

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