Problem 65 - Physics for Scientists and Engineers with Modern Physics

Question

What fraction of the speed of light \(c\) is the speed of a \(7.0-\mathrm{TeV}\) proton?

Step-by-Step Solution

Answer

The speed of a 7.0-TeV proton is about 0.9999999 times the speed of light \(c\).

Step 1: Identify the given values

We are given the energy of the proton, which is 7.0 TeV. To solve the problem, we will convert this energy into joules, since the speed of light, which is the reference speed, is typically measured in meters per second widely recognized in the International System of Units (SI). We know 1 electron-volt (eV) is equal to approximately \(1.602 imes 10^{-19}\) Joules. Therefore, \(1 \text{ TeV} = 1 \times 10^{12} \text{ eV} = 1.602 \times 10^{-7} \text{ J} \).

Step 2: Convert the energy to joules

Convert 7.0 TeV to Joules:\[ \text{Energy in Joules} = 7.0 \times 10^{12} \times 1.602 \times 10^{-19} = 1.1214 \times 10^{-6} \text{ J} \]

Step 3: Relate energy to velocity

For relativistic speeds, use the formula for relativistic kinetic energy: \(E = \gamma m c^2 - m c^2\), where \(\gamma\) is the Lorentz factor given by \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \). However, since \(E = \gamma m c^2 - m c^2 = (\gamma - 1)mc^2\), and \(E\) is considerably higher than the rest energy (7.0 TeV is much greater than the proton's rest energy of about 1 GeV = 0.001 TeV), we can simplify calculations.

Step 4: Approximate using the relativistic equation

At very high energies such as 7.0 TeV, \(\gamma \approx \frac{E}{mc^2} = \frac{7 \times 10^{12} \times 1.602 \times 10^{-19}}{1.67 \times 10^{-27} \times (3.0 \times 10^8)^2} \), where proton mass \(m \approx 1.67 \times 10^{-27}\, \text{kg}\).

Step 5: Compute \(\gamma\) and velocity \(v\)

Let's calculate \(\gamma\):\[\gamma \approx \frac{1.1214 \times 10^{-6}}{1.67 \times 10^{-27} \times 9.0 \times 10^{16}} \approx 7463\]For a very high \(\gamma\), \(1 - \frac{v^2}{c^2} \approx \frac{1}{2\gamma^2}\), so \(\frac{v}{c} \approx 1 - \frac{1}{2\gamma^2}\).

Step 6: Final velocity fraction calculation

Calculate the fraction of \(v\) in terms of \(c\):\[ \frac{v}{c} \approx 1 - \frac{1}{2 \times (7463)^2} \approx 1 - \frac{1}{2 \times 55683369} \]Finally, clean numerical computations show \(\frac{v}{c} \approx 1 - 0.0000001\approx 0.9999999\).

Key Concepts

Relativistic Kinetic Energy

In the realm of relativistic physics, kinetic energy isn't as straightforward as in classical mechanics. At high velocities approaching the speed of light, the formula for kinetic energy changes significantly. Instead of using \( KE = \frac{1}{2}mv^2 \), we must use the relativistic kinetic energy formula: \( KE = (\gamma - 1)mc^2 \), where \( \gamma \) is the Lorentz factor and \( m \) is the rest mass of the object.
This formula accounts for the fact that as an object's speed approaches the speed of light, its energy doesn't just increase linearly. Instead, it becomes much more significant due to the effects of Einstein's theory of special relativity.

• The term \( mc^2 \) represents the rest energy of the object, which is the energy an object possesses even when at rest.
• The factor \( \gamma - 1 \) represents the increase in energy due to the object’s movement.

When dealing with particles like protons at high energies, such as 7.0 TeV, relativistic kinetic energy provides crucial insights into how these particles behave under extreme conditions.

Lorentz Factor

The Lorentz factor, denoted by \( \gamma \), is a key concept in understanding relativistic speeds. It quantifies the amount by which time, length, and relativistic mass undergo change due to the velocity of an object relative to the speed of light. The formula for the Lorentz factor is: \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]

• As the velocity \( v \) approaches the speed of light \( c \), \( \gamma \) becomes very large.
• This indicates significant relativistic effects, altering the measurements of time and space.

In our exercise, a proton with an energy of 7 TeV, much higher than its rest energy, means that the Lorentz factor is significantly greater than 1, illustrating how velocities close to \( c \) considerably affect its physical properties.

Speed of Light

In the universe of physics, the speed of light, denoted as \( c \), is a fundamental constant and the ultimate speed limit. It is approximately 3.0 x 10^8 meters per second. This speed is so critical because it represents the maximum speed at which information or matter can travel through the vacuum of space.

• Nothing with mass can reach, or exceed, \( c \). However, objects can get arbitrarily close at relativistic speeds.
• As speeds approach \( c \), relativistic effects such as time dilation and length contraction become prominent.

In the given problem, we calculated that a 7 TeV proton travels at nearly 99.99999% of the speed of light. This illustrates just how high-energy particles in accelerators approach the universe's speed limit, providing an exceptional understanding of relativistic physics.