The four fundamental equations

Consider a body moving in a straight line with constant acceleration $a$. Between two events A and B, four equations hold, each of which links three of the four kinematic quantities.

Equations of uniformly accelerated motion

$s_B - s_A = v_A\,(t_B - t_A) + \frac{1}{2}\,a\,(t_B - t_A)^2$ $v_B - v_A = a\,(t_B - t_A)$ $\frac{v_A + v_B}{2} = \frac{s_B - s_A}{t_B - t_A}$ $v_B^2 - v_A^2 = 2\,a\,(s_B - s_A)$

The way to use them is always the same: look at which data are known and choose the equation that contains a single unknown. Here is what each one links:

What each equation links

• displacement: links $s$, $v_A$, $a$, $t$;
• velocity: links $v$, $a$, $t$;
• average velocity: links $s$, $v_A$, $v_B$, $t$;
• time-free: links $v$, $a$, $s$ — no $t$!

The last equation, $v_B^2 - v_A^2 = 2a(s_B - s_A)$, is particularly valuable: it is the only one in which time does not appear. When time is not known and is not of interest, it allows velocity, acceleration and displacement to be linked directly without having to calculate time as an intermediate step.

Links

Topics: Kinematics Concepts: Uniformly accelerated motion Skills: Symbolic setup

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