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Chapter 36: Problem 99

(II) For a 1.0 -kg mass, make a plot of the kinetic energy as a function of speed for speeds from 0 to \(0.9 c,\) using both the classical formula \(\left(K=\frac{1}{2} m v^{2}\right)\) and the correct relativistic formula \(\left(K=(\gamma-1) m c^{2}\right).\)

Short Answer

Expert verified
Relativistic KE exceeds classical KE at high speeds.

Step by step solution

01

Understand the Given Formulas

There are two formulas provided for kinetic energy (KE). The classical formula is \( K = \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is the velocity. The relativistic formula is \( K = (\gamma - 1) m c^2 \), where \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) and \( c \) is the speed of light. Our goal is to compare KE from 0 to \( 0.9c \) for a 1.0 kg mass.
02

Calculate Kinetic Energy for Different Speeds

We need to calculate KE using both formulas for speeds ranging from 0 to \( 0.9c \). This typically involves creating a set of speeds (e.g., 0, 0.1c, 0.2c, ..., 0.9c) and applying each formula to find the kinetic energy at each speed.
03

Classical Kinetic Energy Calculation

For each speed \( v \), use the classical formula \( K = \frac{1}{2} m v^2 \). For example, when \( v = 0.1c \), \( K = \frac{1}{2} (1.0) (0.1c)^2 = \frac{1}{2} (1.0) (0.01c^2) \). Repeat this for each speed up to \( 0.9c \).
04

Relativistic Kinetic Energy Calculation

For each speed \( v \), calculate \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \) and then use the relativistic formula \( K = (\gamma - 1) m c^2 \). For example, when \( v = 0.1c \), first calculate \( \gamma = \frac{1}{\sqrt{1 - 0.01}} \). Use this value to calculate \( K \). Repeat for each speed.
05

Plot the Kinetic Energy

With all kinetic energy values calculated, create two plots of kinetic energy against speed. One plot is for the classical KE at \( K = \frac{1}{2} m v^2 \), and the other is for the relativistic KE at \( K = (\gamma - 1) m c^2 \). The x-axis represents speed (in multiples of \( c \)), and the y-axis represents kinetic energy.
06

Analyze the Results

Examine the plots to understand how the classical and relativistic kinetic energies differ. The classical KE increases quadratically, while the relativistic KE prediction diverges at higher speeds due to the \( \gamma \) factor. Notice the significant difference as speed approaches \( 0.9c \).

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Key Concepts

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

Classical Mechanics
The classical mechanics approach provides a simple yet powerful way to understand motion, forces, and energy without involving the intricacies of high velocities. In the realm of classical mechanics, kinetic energy (KE) can be calculated using the formula: \( K = \frac{1}{2} m v^2 \). Here:
  • \( K \) is the kinetic energy,
  • \( m \) is the mass of the object, and
  • \( v \) is the velocity.
This formula works perfectly for everyday speeds but shows inconsistencies when applied to speeds approaching the speed of light. It assumes that mass remains constant regardless of speed, which doesn't hold true under extreme conditions.
Relativistic Mechanics
When objects move at significant fractions of the speed of light, classical mechanics falls short. Relativistic mechanics, birthed from Einstein's theory of relativity, addresses these situations. It adjusts for changes in the object's mass and energy as its speed approaches the speed of light. The formula for relativistic kinetic energy is:
  • \( K = (\gamma - 1) m c^2 \)
This equation incorporates the Lorentz factor, \( \gamma \), instead of assuming a linear increase in kinetic energy. It enhances accuracy by factoring in the relativistic effects that alter the object's mass and energy at high speeds.
Speed of Light
The speed of light, commonly denoted as \( c \), is a fundamental constant in physics, approximately equal to \( 3 \times 10^8 \) meters per second. The limit set by the speed of light has crucial implications in both classical and relativistic mechanics:
  • In classical mechanics, the speed of light is not considered a barrier, leading to incorrect predictions at very high speeds.
  • In relativistic mechanics, reaching or surpassing the speed of light is impossible for anything with mass. As an object's speed nears the speed of light, its kinetic energy increases without bound, approaching infinity.
Thus, the speed of light is a cornerstone in understanding the contrast between classical and relativistic physics.
Lorentz Factor
The Lorentz factor, represented by \( \gamma \), is a pivotal component in relativistic mechanics calculations. It is defined as:
  • \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)
This factor accounts for time dilation and length contraction effects that arise at high velocities. When a particle's speed increases, the Lorentz factor approaches infinity as it nears the speed of light, indicating infinite energy.
Incorporating \( \gamma \), the relativistic kinetic energy becomes accurate even at speeds close to \( c \), marking a significant deviation from classical predictions.
Energy Comparison
Comparing the kinetic energy computed under classical and relativistic frameworks reveals how they diverge at high speeds. At low speeds, both classical and relativistic kinetic energies yield similar results, making classical approaches practical for everyday applications.
However, as the object's speed approaches a substantial fraction of the speed of light, their results differ significantly:
  • Classical predictions continue to increase quadratically and inaccurately suggest an object can surpass the speed of light.
  • Relativistic predictions exponentially increase and predict that immense energy is required to further accelerate the mass.
This comparison underscores the necessity of using relativistic mechanics for high-speed calculations to ensure accurate and feasible outcomes in physics.

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Most popular questions from this chapter

A quasar emits familiar hydrogen lines whose wave-lengths are 7.0\(\%\) longer than what we measure in the laboratory. (a) Using the Doppler formula for light, estimate the speed of this quasar. (b) What result would you obtain if you used the "classical" Doppler shift?

(II) In its own reference frame, a box has the shape of a cube \(2.0 \mathrm{~m}\) on a side. This box is loaded onto the flat floor of a spaceship and the spaceship then flies past us with a horizontal speed of \(0.80 c\). What is the volume of the box as we observe it?

(III) A particle of mass \(m\) is projected horizontally at a relativistic speed \(v_{0}\) in the \(+x\) direction. There is a constant downward force \(F\) acting on the particle. Using the definition of relativistic momentum \(\vec{\mathbf{p}}=\gamma m \vec{\mathbf{v}}\) and Newton's second law \(\vec{\mathbf{F}}=d \vec{\mathbf{p}} / d t,(\) a) show that the \(x\) and \(y\) components of the velocity of the particle at time \(t\) are given by $$\begin{aligned} v_{x}(t) &=p_{0} c /\left(m^{2} c^{2}+p_{0}^{2}+F^{2} t^{2}\right)^{\frac{1}{2}} \\ v_{y}(t) &=-F c t /\left(m^{2} c^{2}+p_{0}^{2}+F^{2} t^{2}\right)^{\frac{1}{2}} \end{aligned}$$ where \(\quad p_{0}\) is the initial momentum of the particle. (b) Assume the particle is an electron \(\left(m=9.11 \times 10^{-31} \mathrm{kg}\right)\) with \(v_{0}=0.50 c \quad\) and \(F=1.00 \times 10^{-15} \mathrm{N} .\) Calculate the values of \(v_{x}\) and \(v_{y}\) of the electron as a function of time \(t\) from \(t=0\) to \(t=5.00 \mu \mathrm{s}\) in intervals of 0.05\(\mu \mathrm{s} .\) Graph the values to show how the velocity components change with time during this interval. (c) Is the path parabolic, as it would be in classical mechanics? Explain.

(II) A friend speeds by you in her spacecraft at a speed of \(0.760 c .\) It is measured in your frame to be \(4.80 \mathrm{~m}\) long and \(1.35 \mathrm{~m}\) high. \((a)\) What will be its length and height at rest? (b) How many seconds elapsed on your friend's watch when 20.0 s passed on yours? \((c)\) How fast did you appear to be traveling according to your friend? \((d)\) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

(a) What is the speed \(v\) of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference \(c-v\). Such speeds are reached in the Stanford Linear Accelerator, SLAC. (b) If the electrons travel in the lab through a tube \(3.0 \mathrm{~km}\) long (as at SLAC), how long is this tube in the electrons' reference frame?
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