In special relativity, energy and momentum are tied together much more tightly than in classical mechanics. A particle
with rest mass \(m\) has a nonzero rest energy even when it is not moving, and as its speed increases, both its total
energy and its momentum grow according to the Lorentz factor \(\gamma\). The calculator uses the standard one-particle
formulas for total energy, rest energy, kinetic energy, and momentum, and it also compares them through the
energy-momentum relation.
The starting point is the speed fraction \(\beta = v/c\), where \(v\) is the particle speed and \(c\) is the speed of
light. From this, the Lorentz factor is:
Lorentz factor. This factor controls how strongly relativistic corrections appear.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}} \\
&= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.
\end{aligned}
\]
When the speed is small compared with \(c\), \(\gamma\) is very close to 1 and the relativistic formulas reduce
approximately to the familiar low-speed results. When the speed approaches \(c\), \(\gamma\) increases sharply, which
makes the total energy and momentum rise very quickly.
Energy formulas
A particle at rest still has energy. That rest energy is
Rest energy. This is the energy associated only with the rest mass.
\[
\begin{aligned}
E_0 &= m c^2.
\end{aligned}
\]
When the particle moves, its total relativistic energy becomes
Total energy. This includes both the rest-energy part and the motion-related part.
\[
\begin{aligned}
E &= \gamma m c^2.
\end{aligned}
\]
The kinetic energy is then the excess above the rest energy:
Kinetic energy. Subtract the rest-energy contribution from the total energy.
\[
\begin{aligned}
K &= E - E_0 \\
&= \gamma m c^2 - m c^2 \\
&= (\gamma - 1)m c^2.
\end{aligned}
\]
This formula is especially useful because it connects directly to the Lorentz factor. At low speed, it approaches the
classical expression \(K \approx \tfrac12 m v^2\), but at relativistic speed it grows much faster.
Momentum formula
Relativistic momentum is not simply \(m v\). The correct formula includes \(\gamma\):
Relativistic momentum. The momentum grows with both speed and the Lorentz factor.
\[
\begin{aligned}
p &= \gamma m v.
\end{aligned}
\]
This form becomes very important in particle accelerators and high-energy physics, where even a small increase in speed
near \(c\) can require a very large increase in momentum and energy.
Energy-momentum relation
One of the most important checks in relativistic mechanics is the exact relation between energy, momentum, and mass:
Energy-momentum relation. This relation stays valid in every inertial frame.
\[
\begin{aligned}
E^2 &= (p c)^2 + (m c^2)^2.
\end{aligned}
\]
This equation is extremely useful because it links all the main relativistic quantities in a single invariant formula.
It is also the reason energy-momentum diagrams are so common in accelerator physics and collision problems.
Worked electron example
Consider an electron with rest mass \(m = 9.11\times 10^{-31}\ \mathrm{kg}\) moving at \(v = 0.99c\). First compute
the Lorentz factor:
Step 1. Compute \(\gamma\) at \(\beta = 0.99\).
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-(0.99)^2}} \\
&= \frac{1}{\sqrt{1-0.9801}} \\
&= \frac{1}{\sqrt{0.0199}} \\
&\approx 7.09.
\end{aligned}
\]
The electron rest energy is about \(0.511\ \mathrm{MeV}\). Using that value:
Step 2. Compute the total energy.
\[
\begin{aligned}
E &= \gamma E_0 \\
&= 7.09 \cdot 0.511\ \mathrm{MeV} \\
&\approx 3.62\ \mathrm{MeV}.
\end{aligned}
\]
Step 3. Compute the kinetic energy.
\[
\begin{aligned}
K &= E - E_0 \\
&= 3.62 - 0.511 \\
&\approx 3.11\ \mathrm{MeV}.
\end{aligned}
\]
Step 4. Use the energy-momentum relation to estimate the momentum.
\[
\begin{aligned}
p c &= \sqrt{E^2 - E_0^2} \\
&= \sqrt{(3.62)^2 - (0.511)^2}\ \mathrm{MeV} \\
&\approx 3.59\ \mathrm{MeV}.
\end{aligned}
\]
So the corresponding momentum is approximately
\[
\begin{aligned}
p &\approx 3.59\ \mathrm{MeV}/c.
\end{aligned}
\]
This example shows a typical relativistic pattern: once the particle speed is already very close to \(c\), large
increases in energy and momentum produce only tiny further increases in speed. That is why accelerators are often
described in terms of energy or momentum rather than raw speed.
How to interpret the calculator outputs
The calculator reports \(E_0\), \(E\), \(K\), and \(p\) separately so their physical meaning stays clear. The rest
energy depends only on the mass, the total energy includes motion, the kinetic energy measures the motion-related part,
and the momentum tracks how difficult the particle is to stop or deflect. The additional energy-momentum relation helps
verify that the computed quantities are internally consistent.
Core summary formulas. These are the main relations used by the calculator.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}}, \\
E_0 &= m c^2, \\
E &= \gamma m c^2, \\
K &= (\gamma - 1)m c^2, \\
p &= \gamma m v, \\
E^2 &= (p c)^2 + (m c^2)^2.
\end{aligned}
\]
At a more advanced university level, these formulas are packaged together using the four-momentum vector. That
framework is the natural next step after mastering the single-particle energy and momentum relations used here.
