Problem 50 - Fundamentals of Physics

Question

To four significant figures, find the following when the kinetic energy is \(10.00 \mathrm{MeV}:\) (a) \(\gamma\) and \((\mathrm{b}) \beta\) for an clectron \(\left (E_{\mathrm{i}}=\right.\) \(0.510998 \mathrm{MeV}\) ), (c) \(\gamma\) and (d) \( \beta\) for a proton \(\left(E_{0}=938.272 \mathrm{MeV}\right)\) and (c) \(\gamma\) and (f) \(\beta\) for an \(\alpha\) particle \(\left (E_{0}=3727.40 \mathrm{McV}\right)\)

Step-by-Step Solution

Answer

For the 10.00 MeV kinetic energy: (a) Electron: \( \gamma \approx 20.57, \beta \approx 0.999 \), (c) Proton: \( \gamma \approx 1.01065, \beta \approx 0.145 \), (e) Alpha particle: \( \gamma \approx 1.00268, \beta \approx 0.0731 \).

Step 1: Understanding the Rest Energy and Kinetic Energy

The rest energy of a particle is the energy it possesses when at rest. It is given by the product of the rest mass and the speed of light squared. The kinetic energy (KE) is the energy the particle has due to its motion. In this problem, we need to find out the values of \( \gamma \) (Lorentz factor) and \( \beta \) (velocity as a fraction of the speed of light) for different particles given a kinetic energy of 10.00 MeV.

Step 2: Calculate Lorentz Factor \( \gamma \) for an Electron

The Lorentz factor \( \gamma \) is calculated using the formula:\[ \gamma = 1 + \frac{K.E.}{E_0} \]where \( K.E. = 10.00 \; \text{MeV} \) and \( E_0 = 0.510998 \; \text{MeV} \). Substitute the given values:\[ \gamma = 1 + \frac{10.00}{0.510998} \approx 20.57 \] to four significant figures.

Step 3: Calculate \( \beta \) for the Electron

Once \( \gamma \) is known, \( \beta \) is given by:\[ \beta = \sqrt{1 - \frac{1}{\gamma^2}} \]Substituting \( \gamma = 20.57 \):\[ \beta = \sqrt{1 - \frac{1}{20.57^2}} \approx 0.999 \] to four significant figures.

Step 4: Calculate Lorentz Factor \( \gamma \) for a Proton

Use the same formula for the proton:\[ \gamma = 1 + \frac{K.E.}{E_0} = 1 + \frac{10.00}{938.272} \approx 1.01065 \] to four significant figures.

Step 5: Calculate \( \beta \) for the Proton

Using the \( \gamma \) value for the proton:\[ \beta = \sqrt{1 - \frac{1}{1.01065^2}} \approx 0.145 \] to four significant figures.

Step 6: Calculate Lorentz Factor \( \gamma \) for an Alpha Particle

For the alpha particle:\[ \gamma = 1 + \frac{K.E.}{E_0} = 1 + \frac{10.00}{3727.40} \approx 1.00268 \] to four significant figures.

Step 7: Calculate \( \beta \) for the Alpha Particle

Using the \( \gamma \) value for the alpha particle:\[ \beta = \sqrt{1 - \frac{1}{1.00268^2}} \approx 0.0731 \] to four significant figures.

Key Concepts

Lorentz Factor

When we talk about the Lorentz factor, we're diving into a fundamental aspect of Einstein's special relativity. The Lorentz factor, denoted as \( \gamma \), is a way to measure time dilation, length contraction, and more importantly, the relation between the rest frame and the moving frame of an object.
In mathematical terms, \( \gamma \) is expressed as:

• \( \gamma = \frac{1}{\sqrt{1 - \beta^2}} \)

where \( \beta \) is the velocity of the object as a fraction of the speed of light \( c \).This equation helps us understand the changes in physical experiences at near-light speeds. In our exercise, we calculate \( \gamma \) by relating kinetic energy (KE) to the rest energy \( E_0 \) using:

• \( \gamma = 1 + \frac{K.E.}{E_0} \)

This relation ties together energy changes as objects approach incredible speeds, getting us closer to the realm of relativistic physics.

Kinetic Energy

Kinetic energy in the realm of special relativity is intriguing because it goes beyond the classical notion of energy carried by motion \((KE = \frac{1}{2}mv^2)\). At speeds approaching that of light, this classic formula falls short. Relativistic kinetic energy takes into account the increase in mass energy as objects accelerate.Special relativity provides this new perspective on kinetic energy with the formula:

• \(K.E. = (\gamma - 1)E_0 \)

This formula shows the energy gained due to motion is significantly different from Newtonian ideas when velocities soar near light speed. The KE assists in calculating the Lorentz factor by unraveling how much energy is initially present \((E_0)\) in its rest state and how that changes as it moves.

Rest Energy

Rest energy is another core concept in special relativity and it's fundamentally tied to the famous equation \( E = mc^2 \) proposed by Einstein. When an object is at rest, it possesses a certain amount of energy purely based on its mass, known as rest energy.To calculate the rest energy \(E_0\), we consider:

• \(E_0 = mc^2 \)

where \( m \) is the rest mass and \( c \) is the speed of light. The equation reveals that even without motion, mass harbors energy just by virtue of existing. In our exercise, the rest energy plays a pivotal role in deriving \( \gamma \) and consequently \( \beta \), as it represents the energy benchmark from which kinetic energy contributions are evaluated.