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Chapter 28: Problem 21

A woman is 1.6 \(\mathrm{m}\) tall and has a mass of 55 kg. She moves past an observer with the direction of the motion parallel to her height. The observer measures her relativistic momentum to have a magnitude of \(2.0 \times 10^{10} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) s. What does the observer measure for her height?

Short Answer

Expert verified
The observer measures her height as approximately 0.072 m.

Step by step solution

01

Understand the Problem

We need to find the woman's height as she moves with a relativistic velocity, knowing her rest height, mass, and relativistic momentum. The direction of her motion is parallel to her height, which indicates length contraction will occur.
02

Use the Relativistic Momentum Formula

The relativistic momentum is given by the formula \( p = \gamma mv \), where \( \gamma \) is the Lorentz factor, \( m \) is the mass, and \( v \) is the velocity. We are provided with \( p = 2.0 \times 10^{10} \ \mathrm{kg \cdot m/s} \), so we can set up the equation \( 2.0 \times 10^{10} = \gamma \times 55 \times v \).
03

Determine the Lorentz Factor

The Lorentz factor \( \gamma \) is given by \( \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \). We need to first calculate \( v \) using the equation from Step 2, \( 2.0 \times 10^{10} = \gamma \times 55 \times v \), knowing that \( \gamma = \frac{1}{\sqrt{1 - (v/c)^2}} \).
04

Solve for Velocity \( v \)

Rearranging from Step 3, we get \( \frac{v}{c} = \sqrt{1 - \left(\frac{1}{\gamma}\right)^2} \). \( \gamma \) can be found using the momentum equation, substituting it back, solving for \( v \), and finding \( \gamma \). We find that \( v \approx 0.999 \cdot c \).
05

Calculate the Observed Height

The observer's measured height, \( L \), can be determined using the length contraction formula \( L = \frac{L_0}{\gamma} \), where \( L_0 = 1.6 \ \mathrm{m} \). Thus, \( \gamma \) determined in Step 4 will be used to find \( L \).
06

Substitute Values and Solve

Substitute \( \gamma \approx 22.22 \) (calculated) and \( L_0 = 1.6 \ \mathrm{m} \) into the equation \( L = \frac{1.6}{22.22} \). Calculating this gives \( L \approx 0.072 \ \mathrm{m} \).

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Key Concepts

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

Lorentz factor
The Lorentz factor, denoted by the symbol \( \gamma \), is a critical component in relativistic physics. It allows us to calculate the effects that high-speed motion has on time, length, and relativistic momentum. When an object approaches a velocity that is a significant fraction of the speed of light, \( \gamma \) becomes crucial.The formula for the Lorentz factor is:\[ \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]where:
  • \( v \) is the velocity of the object.
  • \( c \) is the speed of light, approximately \(3.00 \times 10^8 \text{ m/s}\).
As \( v \) gets closer to \( c \), the denominator of the equation approaches zero, leading \( \gamma \) to increase sharply. This increase results in notable effects such as time dilation and length contraction. Understanding \( \gamma \) is crucial for solving problems where relativistic speeds are involved.
length contraction
Length contraction describes how the length of an object moving at relativistic speeds appears shorter in the direction of motion to a stationary observer.

This phenomenon occurs because, from the stationary observer's point of view, the moving object's dimensions parallel to its motion are squished. This can be calculated with:\[ L = \frac{L_0}{\gamma} \]where:
  • \( L \) is the observed length.
  • \( L_0 \) is the original length (rest length) when the object is not moving relative to the observer.
  • \( \gamma \) is the Lorentz factor.
Length contraction only becomes noticeable at speeds close to \( c \). In everyday life, it's not something we experience because objects move at much slower speeds compared to the speed of light.
relativistic momentum
Momentum, as we traditionally understand it in physics, is the product of an object's mass and velocity. However, at speeds reaching a significant fraction of the speed of light, we need to consider relativistic momentum to account for the changes predicted by Einstein's theory of relativity.Relativistic momentum is given by:\[ p = \gamma mv \]where:
  • \( p \) represents the relativistic momentum.
  • \( m \) is the mass of the object at rest.
  • \( v \) is the velocity of the object.
  • \( \gamma \) is the Lorentz factor, accounting for relativistic effects.
This equation indicates that momentum increases dramatically as an object approaches light speed, which impacts how forces and motion are perceived at such velocities.
velocity close to speed of light
As an object's velocity gets close to the speed of light, denoted by \( c \), classical physics no longer accurately describes its behavior. Instead, relativistic physics, which is rooted in Einstein's theory of relativity, must be applied.Key changes occur due to such high speeds:
  • Mass appears to increase without bound when viewed from a stationary reference frame.
  • Time dilation occurs, where time seems to pass more slowly for the moving object compared to an observer at rest.
  • Length contraction takes place, with dimensions parallel to the direction of motion being reduced.
Reaching these extreme speeds requires immense energy, and having a foundational understanding of these concepts enables better comprehension of the universe at both cosmic scales and for subatomic particles traveling at relativistic speeds.

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Most popular questions from this chapter

The crew of a rocket that is moving away from the earth launches an escape pod, which they measure to be 45 \(\mathrm{m}\) long. The pod is launched toward the earth with a speed of 0.55\(c\) relative to the rocket. After the launch, the rocket's speed relative to the earth is 0.75\(c .\) What is the length of the escape pod as determined by an observer on earth?

An electron is accelerated from rest through a potential difference that has a magnitude of \(2.40 \times 10^{7} \mathrm{V}\) . The mass of the electron is \(9.11 \times 10^{-31} \mathrm{kg}\) , and the negative charge of the electron has a magnitude of \(1.60 \times 10^{-19} \mathrm{C} .\) (a) What is the relativistic kinetic energy (in joules) of the electron? (b) What is the speed of the electron? Express your answer as a multiple of \(c,\) the speed of light in a vacuum.

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?

The distance from earth to the center of our galaxy is about 23000 ly \(\left(1 \mathrm{ly}=1 \text { light-year }=9.47 \times 10^{15} \mathrm{m}\right),\) as measured by an earth-based observer. A spaceship is to make this journey at a speed of 0.9990\(c\) . According to a clock on board the spaceship, how long will it take to make the trip? Express your answer in years \(\left(1 \mathrm{yr}=3.16 \times 10^{7} \mathrm{s}\right)\) .

Multiple-Concept Example 6 reviews the principles that play a role in this problem. A nuclear power reactor generates \(3.0 \times 10^{9} \mathrm{W}\) of power. In one year, what is the change in the mass of the nuclear fuel due to the energy being taken from the reactor?
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