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Chapter 37: Problem 33

A proton (rest mass 1.67 \(\times\) 10\(^{-27}\) kg) has total energy that is 4.00 times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

Short Answer

Expert verified
(a) Kinetic energy: 4.50 脳 10^-10 J, (b) Momentum: 1.90 脳 10^-19 kg m/s, (c) Speed: 2.83 脳 10^8 m/s.

Step by step solution

01

Calculate Rest Energy

The rest energy of a proton is given by the formula \( E_0 = m c^2 \), where \( m = 1.67 \times 10^{-27} \) kg is the rest mass of the proton and \( c = 3 \times 10^8 \) m/s is the speed of light. Calculate \( E_0 \).\[ E_0 = 1.67 \times 10^{-27} \times (3 \times 10^8)^2 \approx 1.50 \times 10^{-10} \text{ J} \]
02

Determine Total Energy

The total energy \( E \) of the proton is given as 4.00 times its rest energy. Thus, \[ E = 4.00 \times E_0 = 4.00 \times 1.50 \times 10^{-10} = 6.00 \times 10^{-10} \text{ J} \]
03

Calculate Kinetic Energy

The kinetic energy \( K \) is the total energy minus the rest energy.\[ K = E - E_0 = 6.00 \times 10^{-10} - 1.50 \times 10^{-10} = 4.50 \times 10^{-10} \text{ J} \]
04

Calculate Momentum

The magnitude of the momentum \( p \) can be found using the formula \( E^2 = (pc)^2 + (E_0)^2 \). Using the given total energy and rest energy, solve for \( p \).\[ (6.00 \times 10^{-10})^2 = (p \times 3 \times 10^8)^2 + (1.50 \times 10^{-10})^2 \]\[ p = \frac{\sqrt{(6.00 \times 10^{-10})^2 - (1.50 \times 10^{-10})^2}}{3 \times 10^8} \approx 1.90 \times 10^{-19} \text{ kg m/s} \]
05

Calculate Speed of Proton

Using the relationship between momentum and kinetic energy for relativistic particles, the speed \( v \) of the proton can be approximated by \( v = \frac{pc}{E} \).\[ v = \frac{1.90 \times 10^{-19} \times 3 \times 10^8}{6.00 \times 10^{-10}} \approx 2.83 \times 10^8 \text{ m/s} \]

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Key Concepts

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

Kinetic Energy
Kinetic energy is the energy a particle possesses due to its motion. In relativistic physics, the kinetic energy is not only dependent on velocity but also on the particle's total energy and rest energy. For a moving proton, the kinetic energy \( K \) can be calculated using the equation \( K = E - E_0 \), where \( E \) is the total energy and \( E_0 \) is the rest energy. Here's what you need to remember:
  • The rest energy \( E_0 \) is the energy the proton would have if it were at rest, calculated by \( E_0 = m c^2 \).
  • Total energy \( E \) includes both rest energy and kinetic energy.
  • Kinetic energy gives us insights into how the energy of a moving proton compares to one that's not moving.
Recognizing the difference between rest and kinetic energy is crucial in understanding how relativistic effects influence high-speed particles.
Rest Energy
Rest energy is a concept that encapsulates the intrinsic energy of a particle at rest. For a proton, this can be found using the formula \( E_0 = m c^2 \), which relates to the mass-energy equivalence principle introduced by Einstein. Points to consider:
  • The rest mass \( m \) is a constant value specific to the particle, like \( 1.67 \times 10^{-27} \) kg for the proton.
  • The speed of light \( c \) is a universally constant value, approximately \( 3 \times 10^8 \) m/s.
  • Rest energy gives the amount of energy contained in the mass of a still particle.
This concept is foundational to understanding how particles behave when subjected to forces and gain motion.
Momentum
Momentum in relativistic physics extends beyond the classical concept of \( p = mv \). Here, it involves the relationship between energy and speed, particularly at high velocities close to the speed of light.The key equation is \( E^2 = (pc)^2 + (E_0)^2 \), where \( E \) is the total energy, \( p \) is the momentum, and \( E_0 \) is the rest energy. To find the momentum, solve for \( p \) by rearranging the formula.Consider these important points:
  • Momentum becomes significant when analyzing particles moving near the speed of light.
  • The relativistic momentum accounts for increases in mass as speed increases.
  • Understanding momentum helps in predicting how particles interact and their paths when external forces are applied.
Grasping relativistic momentum is crucial for analyses in high-energy physics.
Speed of Light
The speed of light \( c \) is not only a key factor in calculating rest energy or momentum but also a crucial constant in understanding relativistic effects. For many problems, such as determining the speed of a proton, it serves as an upper limit for velocity.Highlights include:
  • The speed of light is approximately \( 3 \times 10^8 \) m/s, and it remains consistent across all reference frames.
  • Nothing can travel faster than this speed. So, any computed velocity like \( v = \frac{pc}{E} \) should always be less than \( c \).
  • In relativistic calculations, as objects approach this speed, time and space begin to warp, influencing how we compute energy and momentum.
This concept is not just a measure of speed but an integral part of the framework of modern physics.

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Most popular questions from this chapter

(a) How much work must be done on a particle with mass \(m\) to accelerate it (a) from rest to a speed of 0.090\(c\) and (b) from a speed of 0.900\(c\) to a speed of 0.990\(c\) ? (Express the answers in terms of \(mc^2\).) (c) How do your answers in parts (a) and (b) compare?

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 0.600c. The pursuit ship is traveling at a speed of 0.800c 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?

An alien spacecraft is flying overhead at a great distance as you stand in your backyard. You see its searchlight blink on for 0.150 s. The first officer on the spacecraft measures that the searchlight is on for 12.0 ms. (a) Which of these two measured times is the proper time? (b) What is the speed of the spacecraft relative to the earth, expressed as a fraction of the speed of light \(c\)?

An airplane has a length of 60 m when measured at rest. When the airplane is moving at 180 m/s (400 mph) in the alternate universe, how long would the plane appear to be to a stationary observer? (a) 24 m; (b) 36 m; (c) 48 m; (d) 60 m; (e) 75 m.

Many of the stars in the sky are actually \(binary\space stars\), in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called \(spectroscopic\space binary\space stars. \textbf{Figure P37.68}\) shows the simplest case of a spectroscopic binary star: two identical stars, each with mass \(m\), orbiting their center of mass in a circle of radius \(R\). The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of 4.568110 \(\times\) 10\(^{14}\) Hz. In the light received from the stars by a telescope on the earth, hydrogen light is observed to vary in frequency between 4.567710 \(\times\) 10\(^{14}\) Hz and 4.568910 \(\times\) 10\(^{14}\) Hz. Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (\(Hint\): The speeds involved are much less than \(c\), so you may use the approximate result \(\Delta f/f = u/c\) given in Section 37.6.) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius \(R\) and the mass \(m\) of each star. Give your answer for \(m\) in kilograms and as a multiple of the mass of the sun, 1.99 \(\times\) 10\(^{30}\) kg. Compare the value of \(R\) to the distance from the earth to the sun, 1.50 \(\times\) 10\(^{11}\) m. (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)
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