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Chapter 37: Problem 27

A proton has momentum with magnitude \(p_0\) when its speed is 0.400c. In terms of \(p_0\) , what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

Short Answer

Expert verified
The new momentum is approximately 3.055 times \( p_0 \).

Step by step solution

01

Understand Momentum in Relativity

In relativistic physics, the momentum of a particle is given by the equation \( p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \), where \( m \) is the rest mass of the particle, \( v \) is the velocity, and \( c \) is the speed of light.
02

Express Original Momentum

The given problem states the proton has a momentum \( p_0 \) when its speed \( v_0 \) is 0.400c. Thus, \( p_0 = \frac{mv_0}{\sqrt{1 - \left(\frac{v_0}{c}\right)^2}} = \frac{m \cdot 0.4c}{\sqrt{1 - (0.4)^2}}.\)
03

Introduce New Velocity and Solve for New Momentum

We need to find the momentum when the velocity \( v \) is doubled to 0.800c. Using \( p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \), substitute \( v = 0.8c \) into the equation to get \( p = \frac{m \cdot 0.8c}{\sqrt{1 - (0.8)^2}}.\)
04

Relate the New Momentum to Original Momentum

Calculate the value of \( p \) in terms of \( p_0 \) by substituting \( m \) from \( p_0 \) and simplifying: \( p = \frac{0.8c \cdot p_0 \cdot \sqrt{1 - (0.4)^2}}{0.4c \cdot \sqrt{1 - (0.8)^2}}. \) Cancel terms and evaluate the expression.
05

Simplify the Expression

Compute \( \sqrt{1 - (0.4)^2} = \sqrt{0.84} \) and \( \sqrt{1 - (0.8)^2} = \sqrt{0.36} \). Substituting and simplifying gives \( p = p_0 \cdot \frac{2 \sqrt{0.84}}{\sqrt{0.36}} = p_0 \cdot \frac{2 \sqrt{0.84}}{0.6}. \) Further simplification results in \( p = p_0 \cdot \frac{2 \times 0.9165}{0.6} = p_0 \cdot 3.055.\)

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Key Concepts

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

Proton Speed
In classical mechanics, when we talk about speed, it's often a simple velocity value, such as 20 meters per second. However, when it comes to particles like protons moving at significant fractions of the speed of light, things become a bit more complex. In this context, we refer to this realm as "relativistic speed" because it requires the considerations of Einstein's theory of relativity.
The speed of light, represented by the symbol \( c \), is approximately 299,792,458 meters per second. When a proton moves at 0.400 times the speed of light \( (0.400c) \), it's still considerably slower than light, but fast enough that relativistic effects come into play. Relativistic effects include changes in momentum and time dilation, which aren't considered in classical physics.
Understanding the proton's speed in this context is crucial for correctly applying the relativistic momentum equation and predicting the behavior of subatomic particles in high-energy conditions.
Relativistic Physics
Relativistic physics fundamentally alters the way we understand motion and mass at high speeds. Albert Einstein's theory of relativity introduced concepts that deviate from traditional Newtonian physics, especially when objects approach the speed of light.
In the realm of relativistic physics:
  • Mass and energy are interchangeable, linked by the equation \( E=mc^2 \).
  • Time and space are intertwined, creating what is known as the space-time continuum.
  • As an object's speed approaches the speed of light, its mass appears to increase, requiring more and more energy to continue accelerating.
In this exercise, understanding relativistic physics allows us to calculate the momentum of a proton traveling at significant fractions of the speed of light. These nuanced insights are vital for high-energy physics, such as in the operations of particle accelerators and studying cosmic phenomena.
Momentum Equation
The momentum equation in relativistic physics is quite different from the classical equation \( p = mv \) due to the effects of special relativity. The relativistic momentum equation is given by:
\[ p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}} \]
This formula accounts for the increase in momentum that occurs only at relativistic speeds, where \( v \) approaches \( c \).
When analyzing the problem of finding the momentum of a proton at a doubled speed of 0.800c, this equation helps us quantify how the interaction of velocity and relativistic effects influence momentum. By doubling the proton's speed, we see that its momentum increases not linearly but exponentially due to the relativistic factor \( \sqrt{1 - \frac{v^2}{c^2}} \).
Understanding how to manipulate and apply the relativistic momentum equation enables students to grasp the subtleties of motion at near-light speeds, crucial for any astrophysics or particle physics study.

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Most popular questions from this chapter

As measured by an observer on the earth, a spacecraft runway on earth has a length of 3600 m. (a) What is the length of the runway as measured by a pilot of a spacecraft flying past at a speed of 4.00 \(\times\) 10\(^7\) m/s relative to the earth? (b) An observer on earth measures the time interval from when the spacecraft is directly over one end of the runway until it is directly over the other end. What result does she get? (c) The pilot of the spacecraft measures the time it takes him to travel from one end of the runway to the other end. What value does he get?

Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured in the laboratory?

In certain radioactive beta decay processes, the beta particle (an electron) leaves the atomic nucleus with a speed of 99.95\(\%\) the speed of light relative to the decaying nucleus. If this nucleus is moving at 75.00\(\%\) the speed of light in the laboratory reference frame, find the speed of the emitted electron relative to the laboratory reference frame if the electron is emitted (a) in the same direction that the nucleus is moving and (b) in the opposite direction from the nucleus's velocity. (c) In each case in parts (a) and (b), find the kinetic energy of the electron as measured in (i) the laboratory frame and (ii) the reference frame of the decaying nucleus.

Inside a spaceship flying past the earth at three-fourths the speed of light, a pendulum is swinging. (a) If each swing takes 1.80 s as measured by an astronaut performing an experiment inside the spaceship, how long will the swing take as measured by a person at mission control (on earth) who is watching the experiment? (b) If each swing takes 1.80 s as measured by a person at mission control, how long will it take as measured by the astronaut in the spaceship?

A spacecraft of the Trade Federation flies past the planet Coruscant at a speed of 0.600c. A scientist on Coruscant measures the length of the moving spacecraft to be 74.0 m. The spacecraft later lands on Coruscant, and the same scientist measures the length of the now stationary spacecraft. What value does she get?
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