With Play an object (black: wheel classical, magenta point: relativistic) starts at the origin in x direction with constant acceleration b. Its "classical" worldline is shown in light gray. It would cross the red light cone at x = 2 and achieve speed of light at x = 1 (gradient equals that of the light cone).
At the beginning classical and relativistic objects seem to coincide and to travel along the paraboloid classical worldline, while a red photon runs ahead on its light cone with the speed of light. When speed is no longer small compared to that of light, classical and relativistic worldlines separate. While the "classical", black wheel pursues the gray parabola, the real, magenta colored object draws its relativistic blue worldline (a hyperbola), which finally runs parallel to the light cone when the object is approaching the speed of light.
The difference between "classical" and relativistic case becomes obvious only when the speed is of the order of magnitude of the speed of light. With t scaled in seconds (x = 1 corresponds to 300 000 km), all happenings in "normal life" will be restricted to the immediate neighborhood of the origin.
The gray line is the classical solution for constant acceleration b:
x = 1/2 bt2
derived from the differential equation
d 2x / d t 2= b
The relativistiv movement is numerically calculated with the differential equation
d 2x / d 2t = b sqrt(1-((dx /dt)/c)2) = b sqrt(1-(v(t)/c)2)
The red line is the light cone with x = ct
Play/Pause starts and stops the animation. Reset leads back to the starting condition.
Acceleration can be changed with the slider b. Default value is b = 1 which results in the classical object achieving the speed of light after 1 unit of ct at a distance of 1/2 unit of x km. After 2 ct units the object would surpass the simultaneously started photon.
In scaling it is open which is the unit of time. If time is scaled in seconds, the unit of the x scale is light seconds; if it is scaled in years, it is light years.
The real relativistic path (blue) is a hyperbola, which for small speed is not visibly discernible from the classical parabola. At high speed the hyperbola becomes nearly parallel to the light cone. The object can approach the speed of light, but not achieve it.