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Chapter 20: Problem 79

(a) Find the speed required for the classical and relativistic values of (a) momentum and (b) kinetic energy to differ by \(1 \%\).

Short Answer

Expert verified
The speed at which momentum and kinetic energy differ by 1% is close to but less than the speed of light.

Step by step solution

01

Identify the Formulas

First, we recognize the classical and relativistic formulas for momentum. The classical momentum is given by \( p_{c} = mv \), and the relativistic momentum is \( p_{r} = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \). Similarly, the classical kinetic energy is \( KE_{c} = \frac{1}{2}mv^2 \) and the relativistic kinetic energy is \( KE_{r} = \left( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} - mc^2 \right) \), where \( m \) is the mass of the object, \( v \) is the speed, and \( c \) is the speed of light.
02

Setup the Momentum Equation

We need the conditions where the relativistic and classical momentum differ by 1%. Set up the equation: \( p_{r} = 1.01 \, p_{c} \). Substituting the momentum formulas, we get \( \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} = 1.01 \, mv \).
03

Solve for v in Momentum Equation

From \( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = 1.01 \), square both sides to eliminate the square root: \( \frac{1}{1 - \frac{v^2}{c^2}} = 1.0201 \). This leads to \( 1 - \frac{v^2}{c^2} = \frac{1}{1.0201} \). Solve for \( v^2 \): \( v^2 = c^2(1 - \frac{1}{1.0201}) \). Calculate the value of \( v \).
04

Setup the Kinetic Energy Equation

For kinetic energy to differ by 1%, set up the equation: \( KE_{r} = 1.01 \, KE_{c} \). Substituting the formulas, we have: \( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} - mc^2 = 1.01 \, \frac{1}{2}mv^2 \).
05

Solve for v in Kinetic Energy Equation

Simplify the kinetic energy equation: \( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} - mc^2 = 0.505 \, mv^2 \). Isolate \( v \) by solving the equation \( \frac{c^2}{\sqrt{1 - \frac{v^2}{c^2}}} - c^2 = 0.505 \, v^2 \). Solve this equation for \( v^2 \) and find the value of \( v \).
06

Compare and Conclude

Compare the values of \( v \) obtained for the momentum and kinetic energy conditions. These velocities give us the speed at which relativistic and classical calculations differ by 1%.

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Key Concepts

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

Classical Mechanics
Classical mechanics encompasses the laws and concepts that describe the motion of objects under forces primarily based on Isaac Newton's laws of motion. This framework works well when dealing with everyday speeds and massive astronomical bodies where relativistic effects are negligible.
In classical mechanics, momentum is seen as a product of an object's mass and its velocity:
  • Momentum formula: \( p_c = mv \)
Classical kinetic energy is similarly simplified, calculated using the formula:
  • Kinetic energy: \( KE_c = \frac{1}{2}mv^2 \)
These formulas break down when dealing with objects moving close to the speed of light, which is why we turn to relativistic physics for more accurate predictions.
Kinetic Energy
Kinetic energy refers to the energy that an object possesses due to its motion. In a classical viewpoint, the kinetic energy (KE) of an object is straightforward and increases with the square of its speed:
  • Formula: \( KE_c = \frac{1}{2}mv^2 \)
This relationship holds well at low speeds. However, as speeds approach those of light, classical mechanics no longer provides accurate results, and we must incorporate relativistic effects.
In relativistic physics, the calculation becomes:
  • Relativistic kinetic energy: \( KE_r = \left( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} - mc^2 \right) \)
This ensures that as speed approaches the speed of light \( c \), the kinetic energy increases without bounds, reflecting the near impossibility of reaching or exceeding light speed.
Momentum
Momentum is a measure of the quantity of motion of a moving body and is conserved within an isolated system. It鈥檚 a crucial concept in physics that deals not only with motion but also with collisions and impulse. At classical speeds, momentum is proportional directly to mass and velocity, as given by:
  • Classical momentum: \( p_c = mv \)
For speeds nearing the speed of light, discrepancies arise, and the classical momentum equation no longer suffices for accurate measurement.
The relativistic formulation comes into play with the equation:
  • Relativistic momentum: \( p_r = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \)
This adjustment accounts for the increased momentum an object exhibits as its speed increases to relativistic levels, preventing an object from surpassing the speed of light.
Relativity
Relativity fundamentally transforms our understanding of time, space, and movement. Introduced by Albert Einstein, relativistic physics becomes imperative when objects move at high velocities, approaching the speed of light. Key to this theory is the behavior of time and space at high velocities.
This concept importantly alters our perception of kinetic energy and momentum:
  • High-speed objects exhibit significant increases in kinetic energy and momentum when analyzed using relativistic rather than classical equations.
  • The relativistic transformations prevent objects from reaching or exceeding the speed of light \( c \).
Differences between classical and relativistic calculations grow as velocity increases, leading to more pronounced deviations. This theory shapes modern physics and contributes crucial understanding to a wide range of phenomena, from particles in accelerators to the dynamics of massive cosmic structures.

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Most popular questions from this chapter

A spaceship approaching Earth at \(0.5 c\) fires a particle beam toward Earth. The particles move at \(0.6 c\) relative to the spaceship. At what speed does Earth receive the particles? (a) \(0.75 c\) (b) \(0.8 c^{*},\) (c) \(0.85 c\); (d) 0.90 c.

In the future, speeding tickets for spaceships will be issued based on the spaceship's measured length. The Neptune 3000 spaceship has proper length \(75 \mathrm{~m}\). If the posted speed limit is \(0.6 c,\) will the Neptune 3000 spaceship be ticketed if space police measure its length as \(65 \mathrm{~m} ?\)

A boat moves at \(6.00 \mathrm{~m} / \mathrm{s}\) relative to the water. Find the boat's speed relative to shore when it's traveling (a) downstream and (b) upstream in a river with a \(2.0-\mathrm{m} / \mathrm{s}\) current. (c) The boat travels \(100 \mathrm{~m}\) downstream and then \(100 \mathrm{~m}\) upstream, returning to its original point. Find the time for the round trip, and compare this time with the round-trip time if there were no current. (You can neglect relativity at these slow speeds.)

A spaceship makes the 300 -million-km round-trip journey to the Sun in just \(20 \mathrm{~min}\), as measured by clocks on Earth. How much time elapses on the spaceship's clock? (a) \(20 \mathrm{~min}\) (b) \(17 \mathrm{~min}\) (c) \(14 \mathrm{~min}\); (d) \(11 \mathrm{~min}\).

A spaceship heads away from Earth at \(0.20 c .\) The ship beams 400 -nm light toward Earth. The light strikes a mirror on Earth and is reflected back to the ship. What is the wavelength of light received by the ship? Hint: Find the wavelength of light received at Earth, and then consider that the mirror effectively retransmits light of that wavelength.
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