In classical mechanics, collinear velocities simply add: if one object moves at speed \(u\) inside a frame that is
itself moving at speed \(v\), then an outside observer would expect the total speed to be \(u + v\). Special
relativity changes this rule. Because no physical signal or material object can exceed the speed of light, the correct
composition law must slow down the growth of the result as the speeds approach \(c\). That correction is built into
the relativistic velocity addition formula.
For one-dimensional motion in the same line, the forward composition law is:
Forward addition. Start from an object speed \(u\) measured in a moving frame and a frame speed \(v\) measured in the lab frame.
\[
\begin{aligned}
u' &= \frac{u + v}{1 + \frac{u \cdot v}{c^2}}.
\end{aligned}
\]
Here \(u'\) is the speed measured in the lab frame. The denominator is the crucial relativistic correction. If both
\(u\) and \(v\) are much smaller than \(c\), then the product \(u v / c^2\) is tiny and the denominator is very close
to 1, so the formula reduces almost to the classical sum.
It is often convenient to divide every speed by \(c\). If we define \(\beta_u = u/c\), \(\beta_v = v/c\), and
\(\beta' = u'/c\), then the formula becomes:
Dimensionless form. This is the version used directly by the calculator.
\[
\begin{aligned}
\beta' &= \frac{\beta_u + \beta_v}{1 + \beta_u \cdot \beta_v}.
\end{aligned}
\]
This form is especially useful because it makes the speed-of-light limit easy to see. As long as \(|\beta_u| < 1\)
and \(|\beta_v| < 1\), the result also stays below 1 in magnitude. That means the resulting speed always stays below
\(c\), even when the classical sum would not.
Inverse or backward form
Sometimes you know the observed speed \(u'\) in the lab frame and the frame speed \(v\), and you want to recover the
speed \(u\) measured in the moving frame. Solving the forward equation for \(u\) gives the inverse relation:
Backward recovery. Solve for the unknown speed in the moving frame.
\[
\begin{aligned}
u &= \frac{u' - v}{1 - \frac{u' \cdot v}{c^2}}.
\end{aligned}
\]
In terms of speed fractions, the inverse form is:
\[
\begin{aligned}
\beta_u &= \frac{\beta' - \beta_v}{1 - \beta' \cdot \beta_v}.
\end{aligned}
\]
The signs matter. Positive values represent motion in one chosen direction, and negative values represent motion in
the opposite direction. That is why the same formula can describe both a chase scenario and an opposite-direction
launch.
Worked spaceship example
Suppose a spaceship moves at \(v = 0.8c\) relative to Earth and fires a probe forward at \(u = 0.6c\) relative to
the spaceship. The classical answer would be \(1.4c\), which is impossible. The relativistic formula gives:
Step 1. Build the denominator.
\[
\begin{aligned}
D &= 1 + \beta_u \cdot \beta_v \\
&= 1 + (0.6)(0.8) \\
&= 1 + 0.48 \\
&= 1.48.
\end{aligned}
\]
Step 2. Build the numerator.
\[
\begin{aligned}
N &= \beta_u + \beta_v \\
&= 0.6 + 0.8 \\
&= 1.4.
\end{aligned}
\]
Step 3. Divide numerator by denominator.
\[
\begin{aligned}
\beta' &= \frac{N}{D} \\
&= \frac{1.4}{1.48} \\
&\approx 0.946.
\end{aligned}
\]
So the Earth frame measures the probe speed as about \(0.946c\), not \(1.4c\). This is one of the cleanest examples
of how relativity preserves the speed limit.
Why the result never exceeds c
The denominator grows when the input speeds are both large and in the same direction. That extra denominator factor is
what prevents the result from reaching or exceeding the speed of light. For opposite-direction motion, the numerator
becomes smaller, which can greatly reduce the combined speed and can even reverse the sign depending on which motion is
dominant.
The calculator compares the relativistic result with the classical sum or difference so the contrast is easy to see.
It also includes a visual strip showing the speeds as fractions of \(c\), which makes the light-speed bound more
intuitive.
Connection to more advanced relativity
At a more advanced level, velocity addition is derived from the Lorentz transformation by differentiating the
transformed coordinates. In three dimensions, the addition law becomes more complicated because parallel and
perpendicular components behave differently. This solver stays with the one-dimensional case, which is the standard
entry point for learning the idea.
Core summary formulas. These are the main relations used by the calculator.
\[
\begin{aligned}
u' &= \frac{u + v}{1 + \frac{u \cdot v}{c^2}}, \\
\beta' &= \frac{\beta_u + \beta_v}{1 + \beta_u \cdot \beta_v}, \\
u &= \frac{u' - v}{1 - \frac{u' \cdot v}{c^2}}, \\
\beta_u &= \frac{\beta' - \beta_v}{1 - \beta' \cdot \beta_v}.
\end{aligned}
\]
These relations show the main message of relativistic kinematics: speeds do not add linearly once the speed of light
becomes relevant. Instead, the Lorentz structure of space-time reshapes the addition law so that causality and the
light-speed limit are preserved.
