The Four Kinematics Equations and When to Use Each
Short answer: for motion in a straight line with constant acceleration there are four equations, and each one leaves out one of the five quantities (displacement x, initial velocity v₀, final velocity v, acceleration a, time t). Find the quantity the problem does not give or ask for, and use the equation without it.
The four equations
- v = v₀ + a·t — no displacement x.
- x = v₀·t + ½·a·t² — no final velocity v.
- v² = v₀² + 2·a·x — no time t.
- x = ½·(v₀ + v)·t — no acceleration a.
How to pick one
List the five symbols. Tick the three you know and circle the one you want. The fifth one is missing — choose the equation that does not contain it. Example: a car goes from 36 km/h to rest in 25 m. You know v₀, v and x, you want a, and time is missing, so use v² = v₀² + 2·a·x.
Signs
Pick one direction as positive and keep it. A car slowing down while moving forward has a negative acceleration. A ball thrown upward has a = −9.81 m/s² if up is positive.
Solving on your phone
In Newtora's Constant acceleration topic the first three equations are one tap apart. Choose the equation, tap the unknown, enter the values in any unit, and every step — including the rearranged form and the SI conversion — is shown.
Frequently asked questions
Do these equations work for free fall?
Yes. Free fall is constant acceleration with a = g = 9.81 m/s² downward. Pick a direction as positive and keep the sign consistent.
What if the acceleration is not constant?
Then these four equations do not apply. You need calculus or a graph of velocity against time.
Why did I get a negative time?
Usually a sign problem: a decelerating object needs a negative acceleration if its velocity is positive.