The tetrahedron of equations

There is an elegant way to remember that the four quantities of motion ($s$, $v$, $t$, $a$) and the four equations that link them form a single structure: the tetrahedron of equations. The four vertices are the kinematic quantities; each of the four faces carries one of the four equations of uniformly accelerated motion, and each face involves exactly the three vertices it touches.

The tetrahedron of the equations of motion. On the left, top view: vertex $t$ is the apex at the centre, while $s$, $v$, $a$ form the base; the three lateral faces carry the three equations that contain time. On the right, bottom view: the base face $\{s,v,a\}$ is seen, and at its centre, the only equation that does not contain $t$ (the “time-free equation” $v_B^2 - v_A^2 = 2a\Delta s$), right below the point where $t$ is hidden behind the face.

The image captures a deep symmetry: three equations “see” time (the faces that touch the apex $t$), only one does without it (the base face, opposite $t$). It is no coincidence that the time-free equation is precisely the one at the bottom, far from the time vertex.

Links

Topics: Kinematics Concepts: Uniformly accelerated motion

Related exercises: Carlo’s journey · Stopping distance (reaction and braking) · Proof of the time-free equation