Question

The relativistic kinetic energy of a particle is \(T=(\gamma-1) m c^{2} .\) Use the binomial series to express \(T\) as a series in powers of \(\beta=v / c .\) (a) Verify that the first term is just the non relativistic kinetic energy, and show that to lowest order in \(\beta\) the difference between the relativistic and non relativistic kinetic energies is \(3 \beta^{4} m c^{2} / 8 .\) (b) Use this result to find the maximum speed at which the non relativistic value is within \(1 \%\) of the correct relativistic value.

Short answer

(a) Difference is \(\frac{3\beta^4 mc^2}{8}\). (b) Maximum speed is approximately \(0.1155c\).

Step-by-step solution

Understanding the Formula for Kinetic Energy

The relativistic kinetic energy of a particle is given by \( T = (abla -1)mc^2 \), where \( abla = \frac{1}{\sqrt{1-\beta^2}} \) and \( \beta = \frac{v}{c} \) is the velocity of the particle as a fraction of the speed of light.

Expand Using Binomial Series

We're required to express \( T \) in power series of \( \beta \). Using the binomial series expansion \( (1-x)^{-n} \approx 1 + nx + \frac{n(n+1)}{2}x^2 + \ldots \) for small \( x \), expand abla: \[ abla = (1-\beta^2)^{-1/2} \approx 1 + \frac{1}{2}\beta^2 + \frac{3}{8}\beta^4 + \cdots \].

Calculate Relativistic Kinetic Energy

Substitute the series expansion of \( abla \) into the kinetic energy formula: \[ T = \left( \frac{1}{2}\beta^2 + \frac{3}{8}\beta^4 + \cdots \right)mc^2. \] The first term, \( \frac{1}{2}mv^2 \), is the non-relativistic kinetic energy.

Find the Difference between Relativistic and Non-Relativistic Kinetic Energy

To find the difference to the lowest order in \( \beta \), subtract the non-relativistic term from the relativistic series: \[ T_{rel} - T_{non-rel} = \frac{3}{8}mc^2\beta^4. \] Thus, the difference is \( \frac{3\beta^4 mc^2}{8} \) as required.

Determine Maximum Speed for 1% Error

We want the difference \( \frac{3}{8}mc^2\beta^4 \) to be less than or equal to 1% of \( T_{non-rel} = \frac{1}{2}mv^2 \). Set up the inequality: \[ \frac{3}{8}mc^2\beta^4 \leq \frac{1}{100} \cdot \frac{1}{2}mv^2. \] Simplify and solve for \( \beta \).

Solve the Inequality for \(\beta\)

Simplifying the inequality \[ \frac{3}{8}mc^2\beta^4 \leq \frac{1}{200}mv^2 \], and recall \( v = \beta c \). Substituting back, \[ \frac{3}{8}\beta^4 \leq \frac{1}{200}\beta^2 \]. Further simplifying gives \[ \beta^2 \leq \frac{1}{75} \] resulting in \[ \beta \leq \sqrt{\frac{1}{75}} \approx 0.1155 \].

Key concepts

Binomial Series

The binomial series is a powerful mathematical tool used for approximating complicated expressions. It is especially useful when dealing with terms raised to non-integer powers. A key aspect of the binomial series is its ability to expand expressions of the form \[(1-x)^{-n} \approx 1 + nx + \frac{n(n+1)}{2}x^2 + \ldots\]for small values of \(x\). This allows us to approximate expressions that would otherwise be difficult to compute directly. In this exercise, the relativistic factor \(\gamma\), which is \((1-\beta^2)^{-1/2}\), is expanded using this series.
The expansion reveals corrections to the non-relativistic kinetic energy, enabling us to understand how relativistic effects become significant at high velocities.
These expansions provide not just first-order corrections but also higher-order terms that refine our understanding of particle behavior near the speed of light.

Non-Relativistic Kinetic Energy

Non-relativistic kinetic energy is a classical concept that is calculated with the formula\[ T_{non-rel} = \frac{1}{2}mv^2, \]where \(m\) is the mass and \(v\) is the velocity of an object. This equation is derived under the assumption that the velocity \(v\) is much less than the speed of light \(c\). At such low velocities, relativistic effects, which arise in special relativity, are negligible.
This formula is traditionally used in everyday scenarios, such as calculating kinetic energy for cars or baseballs, where speeds are well within the realm where Newtonian physics applies.
In this exercise, we see how the first term from the binomial expansion corresponds to this non-relativistic kinetic energy, illustrating that it serves as a baseline in relativistic mechanics. As the speed increases and approaches significant fractions of the speed of light, corrections become necessary.

Velocity as a Fraction of the Speed of Light

Velocity as a fraction of the speed of light, often denoted as \(\beta = \frac{v}{c}\), is a dimensionless quantity used in relativity. It simplifies calculations and allows physicists to easily compare speeds to the speed of light \(c\).
When \(\beta\) is small, meaning the speed \(v\) is much less than the speed of light, the relativistic effects are minimal and results closely align with classical Newtonian physics. As \(\beta\) nears 1, we approach the realm where relativistic physics must be applied, due to significant deviations from non-relativistic predictions.
In solving problems that involve high velocities, using \(\beta\) enables a clear understanding of how close particles are to reaching light-speed, which dictates significant physical behavior changes due to relativity.

• For \(\beta \leq 0.1155\), the deviation from non-relativistic kinetic energy remains less than 1%. This illustrates a practical speed limit in situations requiring precision without full relativistic treatment, such as some engineering applications or certain high-speed physics experiments.