Question

Find the speed parameter \(\beta\) and the Lorentz factor \(\gamma\) for a particle whose kinetic energy is \(10 \mathrm{MeV}\) if the particle is \((a)\) an electron, \((b)\) a proton, and \((c)\) an alpha particle.

Short answer

The Lorentz factor and speed parameter for the three particles are calculated individually by using their respective rest masses. They are calculated using the formulas for kinetic energy and speed parameter in special relativity.

Step-by-step solution

Determine the Rest Mass

Determine the rest mass for each particle. The rest mass of an electron (me) is approximately \(0.511 \mathrm{MeV}/c^2\), of a proton (mp) is about \(938.27 \mathrm{MeV}/c^2\), and of an alpha particle (mα) is about \(3727.4 \mathrm{MeV}/c^2\).

Calculate the Lorentz Factor

Calculate the Lorentz factor \(\gamma\) for each particle using the kinetic energy formula. Inversely, the formula \(\gamma = \frac{KE}{mc^2} + 1\) can be used. KE is given as \(10 \mathrm{MeV}\). Plug the values into the formula to get the value of \(\gamma\).

Calculate the Speed Parameter

Calculate the speed parameter \(\beta\) using the formula \(\beta^2 = 1 - \frac{1}{\gamma^2}\). After calculating the Lorentz factor in the previous step, substitute \(\gamma\) in the formula to get \(\beta\).

Key concepts

Kinetic Energy

Kinetic energy in relativistic mechanics is slightly different from classical mechanics. It takes into account the effects of relativity, especially when particles travel at speeds close to the speed of light. In this context, the standard formula for kinetic energy is modified. The relativistic kinetic energy (\( KE \)) of a particle is given by the equation:

\[KE = (\gamma - 1) m c^2\]
Here, \( \gamma \) is the Lorentz factor, \( m \) is the rest mass, and \( c \) is the speed of light. This formula accounts for how energy increases significantly as a particle's speed approaches light speed.

The inclusion of the Lorentz factor in kinetic energy formulas reflects how energy isn't linear as speed increases. Instead, it's influenced by relativistic effects, showing why the study of particles moving close to light speed requires a more complex approach than classical mechanics.

Lorentz Factor

The Lorentz factor is crucial when dealing with high-speed particles. It is defined as:

\[\gamma = \frac{1}{\sqrt{1-\beta^2}}\]
Here, \( \beta = \frac{v}{c} \) is the speed parameter. The Lorentz factor describes how much time dilation, length contraction, and relativistic mass increase occur as an object's speed approaches the speed of light.

• When \( \beta = 0 \) (stationary object), \( \gamma = 1 \). No relativistic effects are observed.
• As \( \beta \) approaches 1, \( \gamma \) increases significantly, enhancing relativistic effects.

The Lorentz factor is essential for calculating relativistic kinetic energy and helps in understanding how particles behave at extremely high speeds. It provides a way to measure just how much the laws of physics adjust compared to everyday experiences.

Rest Mass

Rest mass, often denoted as \( m_0 \), is the mass of a particle when it is at rest. It's an intrinsic property of the particle and remains constant regardless of its speed. In relativistic contexts, rest mass is crucial because it forms the basis for calculating other properties.

Rest mass is used in the relativistic energy and kinetic energy formulas. For instance:

\[E = \gamma m_0 c^2\]
Here, \( E \) represents the total energy of the particle. In the original exercise, the rest masses allow for the calculation of the Lorentz factor and kinetic energy of each particle type (electron, proton, alpha particle). Each has a different rest mass and thus behaves differently under the same kinetic energy conditions.

Speed Parameter

The speed parameter \( \beta \) is defined as the ratio of a particle's velocity to the speed of light:

\[\beta = \frac{v}{c}\]
\( \beta \) is dimensionless and typically ranges from 0 to nearly 1 for many particles studied in relativistic mechanics. A \( \beta \) value of 1 implies that the particle is moving at the speed of light, which is impossible for particles with mass.

• As \( \beta \) increases, relativistic effects become more pronounced.
• In calculations, knowing \( \beta \) aids in determining the Lorentz factor, \( \gamma \), which further helps compute kinetic energy.

Understanding \( \beta \) helps clarify how particles behave under high-speed conditions, making it an essential part of analyzing and predicting particle behavior in relativistic mechanics.