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Chapter 28: Problem 22

Three particles are listed in the table. The mass and speed of each particle are given as multiples of the variables \(m\) and \(v\), which have the values \(m=1.20 \times 10^{-8} \mathrm{kg}\) and \(v=0.200 \mathrm{c} .\) The speed of light in a vacuum is \(c=3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\). Determine the momentum for each particle according to special relativity. $$ \begin{array}{clc} \hline \text { Particle } & \text { Mass } & \text { Speed } \\ \hline \mathrm{a} & m & v \\ \mathrm{b} & \frac{1}{2} m & 2 v \\ \mathrm{c} & \frac{1}{4} m & 4 v \\ \hline \end{array} $$

Short Answer

Expert verified
Particle a: 7.345e-9 kg m/s, Particle b: 7.854e-9 kg m/s, Particle c: 1.200e-8 kg m/s.

Step by step solution

01

Determine relativistic momentum formula

The relativistic momentum \( p \) of a particle is given by the formula: \[ p = \gamma m v \] where \( \gamma \) is the Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), \( m \) is the rest mass of the particle, and \( v \) is its velocity.
02

Calculate Lorentz factor for each particle

For particle a: \[ \gamma_a = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} = \frac{1}{\sqrt{1 - \left(\frac{0.200c}{c}\right)^2}} = \frac{1}{\sqrt{1 - 0.04}} = \frac{1}{\sqrt{0.96}} = 1.0204 \]For particle b: \[ \gamma_b = \frac{1}{\sqrt{1 - \left(\frac{2v}{c}\right)^2}} = \frac{1}{\sqrt{1 - \left(\frac{0.400c}{c}\right)^2}} = \frac{1}{\sqrt{1 - 0.16}} = \frac{1}{\sqrt{0.84}} = 1.0911 \]For particle c: \[ \gamma_c = \frac{1}{\sqrt{1 - \left(\frac{4v}{c}\right)^2}} = \frac{1}{\sqrt{1 - \left(\frac{0.800c}{c}\right)^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} = 1.6667 \]
03

Calculate momentum for each particle

Using the given masses and velocities, calculate the momentum:For particle a: \[ p_a = \gamma_a \cdot m \cdot v = 1.0204 \cdot 1.20 \times 10^{-8} \mathrm{kg} \cdot 0.200 \cdot 3.00 \times 10^8 \mathrm{m/s} \]\[ p_a = 7.345 \times 10^{-9} \mathrm{kg} \cdot \mathrm{m/s} \]For particle b: \[ p_b = \gamma_b \cdot \frac{1}{2}m \cdot 2v = 1.0911 \cdot 0.60 \times 10^{-8} \mathrm{kg} \cdot 0.400 \cdot 3.00 \times 10^8 \mathrm{m/s} \]\[ p_b = 7.854 \times 10^{-9} \mathrm{kg} \cdot \mathrm{m/s} \]For particle c: \[ p_c = \gamma_c \cdot \frac{1}{4}m \cdot 4v = 1.6667 \cdot 0.30 \times 10^{-8} \mathrm{kg} \cdot 0.800 \cdot 3.00 \times 10^8 \mathrm{m/s} \]\[ p_c = 1.200 \times 10^{-8} \mathrm{kg} \cdot \mathrm{m/s} \]

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Key Concepts

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

Relativistic Momentum
In classical physics, momentum is simply the product of mass and velocity: \( p = mv \). However, when dealing with speeds close to the speed of light, special relativity must be considered. Here, momentum changes due to relativistic effects.

The formula for relativistic momentum is given by \( p = \gamma m v \), where \( \gamma \) is the Lorentz factor. This equation shows that momentum increases with speed in a nonlinear fashion, making it different from classical momentum.

Relativistic momentum is crucial when particles move near or at relativistic speeds, providing more accurate results.
Lorentz Factor
The Lorentz factor \( \gamma \) is a key concept in special relativity. It's defined as \( \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \), where \( v \) is the velocity of the particle, and \( c \) is the speed of light.

This factor accounts for time dilation and length contraction, effects that only become significant as an object's speed approaches the speed of light.
  • When \( v \) is much smaller than \( c \), \( \gamma \) is approximately 1, and relativistic effects are negligible.
  • As \( v \) approaches \( c \), \( \gamma \) increases dramatically, indicating stronger relativistic effects.
  • At \( v = c \), \( \gamma \) theoretically becomes infinite, showing the impossibility of reaching light speed for objects with mass.
Understanding how \( \gamma \) modifies quantities like time, length, and momentum helps explain seemingly bizarre relativistic phenomena.
Speed of Light
The speed of light, denoted by \( c \), is a fundamental constant in physics, valued at approximately \( 3.00 \times 10^8 \) m/s. It represents the ultimate speed limit in the universe, a cornerstone of Einstein's theory of special relativity.

According to special relativity:
  • Nothing with mass can reach or exceed the speed of light as it would require infinite energy.
  • Light travels at the same speed in any inertial frame, meaning it's constant regardless of the observer's motion.
  • This constancy leads to relativistic effects like time dilation and length contraction, dramatically altering our perception of time and space.
The concept of \( c \) challenges everyday intuitions but is essential for understanding the universe at high velocities.

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Most popular questions from this chapter

A \(6.00-\mathrm{kg}\) object oscillates back and forth at the end of a spring whose spring constant is \(76.0 \mathrm{N} / \mathrm{m}\). An observer is traveling at a speed of \(1.90 \times 10^{8} \mathrm{m} / \mathrm{s}\) relative to the fixed end of the spring. What does this observer measure for the period of oscillation?

Two twins who are 19.0 years of age leave the earth and travel to a distant planet 12.0 light-years away. Assume that the planet and earth are at rest with respect to each other. The twins depart at the same time on different spaceships. One twin travels at a speed of \(0.900 c,\) and the other twin travels at \(0.500 c .\) (a) According to the theory of special relativity, what is the difference between their ages when they meet again at the earliest possible time? (b) Which twin is older?

How fast must a meter stick be moving if its length is observed to shrink to one-half of a meter?

A rectangle has the dimensions of \(3.0 \mathrm{m} \times 2.0 \mathrm{m}\) when viewed by someone at rest with respect to it. When you move past the rectangle along one of its sides, the rectangle looks like a square. What dimensions do you observe when you move at the same speed along the adjacent side of the rectangle?

There are many astonishing consequences of special relativity, two of which are time dilation and length contraction. Problem 50 reviews these important concepts in the context of a golf game in a hypothetical world where the speed of light is only a little faster than that of a golf cart. Other important consequences of special relativity are the equivalence of mass and energy, and the dependence of kinetic energy on the total energy and on the rest energy. Problem 51 serves as a review of the roles played by mass and energy in special relativity. Imagine playing golf in a world where the speed of light is only \(c=3.40 \mathrm{m} / \mathrm{s} .\) Golfer A drives a ball down a flat horizontal fairway for a distance that he measures as \(75.0 \mathrm{m}\). Golfer \(\mathrm{B}\), riding in a cart, happens to pass by just as the ball is hit (see the figure). Golfer A stands at the tee and watches while golfer \(\mathrm{B}\) moves down the fairway toward the ball at a constant speed of \(2.80 \mathrm{m} / \mathrm{s}\). Concepts: (i) Who measures the proper length of the drive, and who measures the contracted length? (ii) Who measures the proper time interval, and who measures the dilated time interval? Calculations: (a) How far is the ball hit according to golfer \(B ?\) (b) According to each golfer, how much time does it take golfer \(\mathrm{B}\) to reach the ball?
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