How to Choose the Right Constant-Acceleration Equation

The equations require uniform acceleration

Constant-acceleration equations connect position, velocity, acceleration, and time only when acceleration is uniform over the interval being modeled.

A common set is

$$v=v_0+at,$$ $$\Delta x=v_0t+\frac12at^2,$$ $$v^2=v_0^2+2a\Delta x,$$

and, for constant acceleration,

$$\bar v=\frac{v_0+v}{2},\qquad \Delta x=\bar vt.$$

Pick the equation by the missing variable

List the quantities you know and the one you want. Then choose an equation that does not contain an additional unknown.

• If displacement is irrelevant or unknown, $v=v_0+at$ is often useful.
• If time is unknown, $v^2=v_0^2+2a\Delta x$ can eliminate it.
• If final velocity is not needed, $\Delta x=v_0t+\tfrac12at^2$ may be simplest.

Fresh example: stopping distance without time

A car travels at $20\,\mathrm{m/s}$ and brakes with constant acceleration $-5.0\,\mathrm{m/s^2}$. How far does it travel before stopping?

Here $v_0=20\,\mathrm{m/s}$, $v=0$, and time is not given. Use

$$v^2=v_0^2+2a\Delta x.$$

Then

$$0=(20)^2+2(-5.0)\Delta x,$$

so

$$\Delta x=40\,\text{m}.$$

Check the model before using the formulas

These equations are not generally valid over an interval where acceleration changes significantly. If acceleration varies with time or position, divide the motion into justified intervals or use calculus-based relations instead of treating one constant value as exact.

A consistent sign convention also matters: choose a positive direction before substituting velocities and acceleration.