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Chapter 37: Problem 32

If a muon is traveling at \(0.999 c,\) what are its momentum and kinetic energy? (The mass of such a muon at rest in the laboratory is 207 times the electron mass.)

Short Answer

Expert verified
Based on the calculations done in Steps 2 and 3, the relativistic momentum \(p\) and the kinetic energy \(K\) of the muon would be determined.

Step by step solution

01

Calculate the Relative Mass and Velocity

First, calculate the relative mass of the muon with respect to the electron mass. Given that the muon mass is 207 times the electron mass and the electron mass is \(9.10938356 \times 10^{-31}\) kilograms, the approximate mass of the muon is approximately \(1.8757 \times 10^{-28}\) kg. As the velocity of muon, \(v= 0.999c\), where \(c\) is the speed of light or \(3\times10^8\) meters per second, the exact velocity of the muon would be \(v= 0.999 \times 3 \times 10^8 = 299700000\) m/s.
02

Calculate the Relativistic Momentum

The relativistic momentum of an object can be determined using the equation \(p = \frac{mv}{\sqrt{1 - v^2/c^2}}\). Substituting the values \(m =1.8757 \times 10^{-28}\) kg and \(v = 299700000\) m/s into the equation would yield the momentum \(p\).
03

Calculate the Kinetic Energy

For a particle moving at relativistic speed, the kinetic energy \(K\) is given by the equation \(K = mc^2(\frac{1}{\sqrt{1 - v^2/c^2}} - 1)\). After substituting the correct values for \(m\), \(v\), and \(c\), one can calculate the kinetic energy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relativistic Momentum
Understanding the momentum of particles like muons when they approach the speed of light requires the principles of relativistic mechanics. Unlike classical momentum, which is the product of mass and velocity (\( p = mv \)), relativistic momentum takes into account the effects of Einstein's theory of relativity.

Relativistic momentum is given by the formula \( p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( m \) is the rest mass of the particle, \( v \) is its velocity, and \( c \) represents the speed of light in a vacuum. Notice as the velocity of a particle approaches the speed of light, the denominator approaches zero, causing the momentum to dramatically increase.

For our muon traveling at 0.999 times the speed of light, calculating the relativistic momentum involves substituting its rest mass and velocity into the relativistic formula. The resulting value is much higher than what would be predicted by classical physics alone, illustrating the necessity of relativistic equations at high speeds.
Kinetic Energy Calculation
When we talk about kinetic energy in the context of relativity, the calculations tend to be a bit more complex than the classical \( \frac{1}{2}mv^2 \). The formula for relativistic kinetic energy is \( K = mc^2(\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1) \), which starts to significantly diverge from classical kinetic energy as objects move close to the speed of light.

In the case of our muon, we include the mass \( m \), the speed of light \( c \), and the muon's velocity \( v \), then calculate the factor that accounts for the effects of special relativity. The subtraction by 1 ensures we are calculating the increase in energy relative to the rest mass energy of the muon. Substituting the appropriate numerical values gives the surprisingly high kinetic energy, demonstrating how energy increases as an object speeds up towards relativistic velocities.
Relativistic Mass
The concept of 'relativistic mass' stems from the fact that as an object's speed nears the speed of light, its mass seems to increase from the perspective of a stationary observer.

This mass increase isn't due to an actual increase in material or matter, but rather an increase in energy required to accelerate the object as it moves faster. At speeds close to \( c \) (the speed of light), you need a tremendous amount of energy to accelerate an already high-velocity object slightly more. This additional energy behaves as if it adds 'mass' to the object.

For our high-speed muon, the relativistic mass would be significantly larger than its rest mass and can be calculated using the Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). By multiplying the muon's rest mass by the Lorentz factor, we find the effective 'relativistic mass' at its high velocity鈥攁 critical component for understanding how subatomic particles behave in accelerators or cosmic rays.

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Most popular questions from this chapter

A spaceship moving at constant speed \(u\) relative to us broad. casts a radio signal at constant frequency \(f_{0}\). As the spaceship approaches us, we receive a higher frequency \(f\); after it has passed, we receive a lower frequency. (a) As the spaceship passes by, so it is instantaneously moving neither toward nor away from us, show that the frequency we receive is not \(f_{0},\) and derive an expression for the frequency we do receive. Is the frequency we receive higher or lower than \(f_{0} ?\) (Hint: In this case, successive wave crests move the same distance to the observer and so they have the same transit time. Thus \(f\) equals \(1 / T .\) Use the dilation formula to relate the periods in the stationary and moving frames.) (b) A spaceship emits electromagnetic waves of frequency \(f_{0}=345 \mathrm{MHz}\) as measured in a frame moving with the ship. The spaceship is moving at a constant speed \(0.758 c\) relative to us. What frequency \(f\) do we receive when the spaceship is approaching us? When it is moving away? In each case what is the shift in frequency, \(f-f_{0} ?\) (c) Use the result of part (a) to calculate the frequency \(f\) and the frequency shift \(\left(f-f_{0}\right)\) we receive at the instant that the ship passes by us. How does the shift in frequency calculated here compare to the shifts calculated in part (b)?

In the earth's rest frame, two protons are moving away from each other at equal speed. In the frame of each proton, the other proton has a speed of \(0.700 c\). What does an observer in the rest frame of the earth measure for the speed of each proton?

Compute the kinetic energy of a proton (mass \(\left.1.67 \times 10^{-27} \mathrm{~kg}\right)\) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) \(8.00 \times 10^{7} \mathrm{~m} / \mathrm{s}\) and (b) \(2.85 \times 10^{8} \mathrm{~m} / \mathrm{s}\)

One way to strictly enforce a speed limit would be to alter the laws of nature. Suppose the speed of light were \(65 \mathrm{mph}\) and your workplace was 30 miles from your home. Assume you travel to work at a typical driving speed of 60 mph. (a) If you drove at that speed for the round trip to and from work, light, how much would your wristwatch lag your kitchen clock each day? (b) Estimate the length of your car. (c) If you were driving at your estimated driving speed, how long would your car be when viewed from the roadside? (d) What would be the speed relative to you of similar cars traveling toward you in the opposite lane with the same ground speed as you? (e) How long would you measure those cars to be? (f) If the total mass of you and your car was \(2000 \mathrm{~kg}\), how much work would be required to get you up to speed? (Note: Your rest mass energy in this world is \(m c^{2}\), where \(c=65\) mph. ) (g) How much work would be required in the real world, where the speed of light is \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s},\) to get you up to speed?

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of \(0.600 c\). The pursuit ship is traveling at a speed of \(0.800 c\) relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the velocity of the cruiser relative to the pursuit ship be directed toward or away from the pursuit ship? (b) What is the speed of the cruiser relative to the pursuit ship?
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