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Chapter 26: Problem 24

How fast must an object travel for its total energy to be (a) \(1 \%\) more than its rest energy and (b) \(99 \%\) more than its rest energy?

Short Answer

Expert verified
a) Approximately 0.14c; b) 0.71c.

Step by step solution

01

Understanding Rest Energy

The rest energy of an object is given by Einstein's famous equation: \[ E_0 = mc^2 \]where \( m \) is the rest mass and \( c \) is the speed of light. The total energy \( E \) of an object moving at a speed \( v \) is given by \[ E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} \] Our task is to determine \( v \) such that the total energy \( E \) is a specified percentage more than the rest energy \( E_0 \).

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Key Concepts

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

Rest Energy
Rest energy is a foundational concept in physics, first articulated by Albert Einstein. It describes the energy contained within a stationary object due to its mass. This is expressed in the equation \( E_0 = mc^2 \), where:
  • \( E_0 \) represents the rest energy.
  • \( m \) is the object's rest mass.
  • \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters per second.
This equation illustrates that even when an object is not moving, it possesses significant energy because of its mass. Rest energy emphasizes the mass-energy equivalence principle, which states that mass can be converted into energy and vice versa. It serves as a basis for understanding the more complex aspects of relativistic physics.
Einstein's Equation
Einstein's equation \( E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} \) explores the relationship between an object's mass \( m \), its speed \( v \), and its total energy \( E \). This equation is pivotal in understanding how energy increases with velocity. Key elements include:
  • Total energy \( E \) includes both the energy due to motion and the rest energy, \( E_0 = mc^2 \).
  • The denominator \( \sqrt{1 - \frac{v^2}{c^2}} \) is known as the Lorentz factor, which accounts for the effects of special relativity as an object's velocity approaches the speed of light.
As \( v \) approaches \( c \), the Lorentz factor grows significantly, causing the total energy to rise exponentially. This mathematical relationship explains why it's impossible to accelerate an object with mass to the speed of light; the energy required would become infinite.
Speed of Light
The speed of light, denoted by \( c \), is a constant essential to the laws of physics. It is the ultimate speed limit in the universe, at which light and all electromagnetic waves travel in a vacuum. Key points about the speed of light include:
  • The value of \( c \) is approximately \( 3 \times 10^8 \) meters per second.
  • Nothing with mass can reach or exceed the speed of light, as dictated by Einstein's theory of relativity.
  • At or near this speed, time dilation and length contraction become significant, impacting how time and space are perceived relative to moving observers.
Understanding the speed of light is critical when dealing with high-speed physics, such as in the study of particles approaching this cosmic speed limit.
Total Energy
Total energy in relativistic context encompasses both rest energy and the energy resulting from motion. When an object moves, particularly at speeds approaching the speed of light, its total energy is no longer just its rest energy. The equation \( E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} \) illustrates how this total energy is calculated. Important aspects include:
  • At speeds much less than \( c \), the difference between total energy and rest energy is negligible.
  • As \( v \) increases and approaches \( c \), the kinetic energy increases, significantly affecting the total energy.
  • The total energy thus takes into account the relativistic effects, making it a more comprehensive measure of an object's energetic state than rest energy alone.
This understanding helps solve problems related to energy increases beyond rest energy, such as determining velocities for specific percentage increases in total energy.

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Most popular questions from this chapter

A spacecraft moves past a student with a relative velocity of \(0.90 c\). If the pilot of the spacecraft observes 10 min to elapse on his watch, how much time has elapsed according to the student's watch?

The proper lifetime of a muon is \(2.20 \mu \mathrm{s}\). If the muon has a lifetime of \(34.8 \mu\) s according to an observer on Earth, what is the muon's speed, expressed as a fraction of \(c\). relative to the observer?

(a) To see that length contraction is negligible at everyday speeds, determine the length contraction \((\Delta L)\) of an automobile \(5.00 \mathrm{~m}\) long when it is traveling at \(100 \mathrm{~km} / \mathrm{h}\). The diameter of an atomic nucleus is on the order of \(10^{-15} \mathrm{~m}\). How does your answer compare to this? (b) Suppose that it is possible to measure a length contraction for this car of \(0.0100 \mathrm{~mm}\) or larger. What minimum car speed would be required to detect this effect? [Hint for part (a): Express the length contraction in terms of \(x=v / c\) and then recall that if \(x \ll 1\), then \(\sqrt{1-x^{2}} \approx 1-\left(x^{2} / 2\right)\)

A speedboat can travel with a speed of \(50 \mathrm{~m} / \mathrm{s}\) in still water. If the boat is in a river that has a flow speed of \(5.0 \mathrm{~m} / \mathrm{s},\) (a) find the maximum and minimum values of the boat's speed relative to an observer on the riverbank. (b) What is the time difference between a downriver trip (with the current) of \(1000 \mathrm{~m}\) and an upriver trip (against the current)?

A small airplane has an airspeed (speed with respect to air) of \(200 \mathrm{~km} / \mathrm{h}\). Find the time for the airplane to travel \(1000 \mathrm{~km}\) if there is (a) no wind, (b) a headwind of \(35 \mathrm{~km} / \mathrm{h},\) and \((\mathrm{c})\) a tailwind of \(35 \mathrm{~km} / \mathrm{h}\).
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