Each kinematic equation leaves out one of the five motion variables, so the right equation is simply the one that omits the variable you neither know nor need. There are four standard equations, five variables (displacement, initial velocity, final velocity, acceleration, and time), and each equation uses exactly four of them. Once you identify which variable is missing from your problem, you have your equation.
The Five Variables
Every kinematics problem involves the same five quantities:
- d: displacement (how far the object moves from its starting point)
- v₀: initial velocity (how fast it’s moving at the start)
- v: final velocity (how fast it’s moving at the end)
- a: acceleration (how quickly the velocity changes)
- t: time (how long the motion lasts)
In any solvable problem you’ll know three of these and need to find a fourth. The fifth variable won’t appear anywhere in the problem, and that’s your clue for picking the right equation.
The Four Equations and What Each One Skips
Here’s the core selection guide. Each equation is listed alongside the one variable it leaves out:
- v = v₀ + at — no displacement (d). Use this when the problem never mentions how far something travels and you don’t need to find it.
- d = ½(v₀ + v)t — no acceleration (a). Use this when acceleration isn’t given and isn’t asked for.
- d = v₀t + ½at² — no final velocity (v). Use this when the problem doesn’t tell you or ask you how fast the object ends up going.
- v² = v₀² + 2ad — no time (t). Use this when the problem never mentions time and you don’t need to find it.
That’s the entire decision. Count what you know, identify what you want, figure out which variable is irrelevant, and grab the equation that skips it.
A Step-by-Step Selection Process
When you’re staring at a word problem, a consistent routine keeps you from guessing at equations. Here’s a reliable workflow:
First, sketch the situation. Even a rough drawing with an arrow showing direction of motion helps you keep signs straight and spot what the problem is really describing. Second, list every known value by symbol. Pull numbers directly from the problem and assign them to d, v₀, v, a, or t. Many problems also contain implied knowns: “starts from rest” means v₀ = 0, “comes to a stop” means v = 0, and “dropped” means v₀ = 0 with a = 9.8 m/s² downward.
Third, identify what the problem asks you to solve for. You should now have three knowns and one unknown, which accounts for four variables. The fifth variable, the one that’s neither given nor requested, tells you which equation to use. Fourth, substitute your values (with units) into that equation and solve. Checking that your units come out correctly is one of the easiest ways to catch algebra mistakes.
Sign Conventions Matter
Before plugging in numbers, pick a positive direction. The standard convention is that rightward and upward are positive, leftward and downward are negative, but you can choose whatever’s convenient as long as you stay consistent throughout the problem. A car braking while moving to the right has a negative acceleration. A ball thrown upward has a positive initial velocity but a negative acceleration (gravity pulls it back down). Getting one sign wrong will flip your answer, so define your convention on paper before you start calculating.
Free-Fall Problems
Free-fall problems use the exact same four equations with one substitution: acceleration equals the gravitational constant, 9.8 m/s² directed downward. If you’ve defined upward as positive, then a = −9.8 m/s². If you’ve defined downward as positive (which is sometimes more convenient for a falling object), then a = +9.8 m/s².
A common free-fall scenario: you drop a ball from a building and want to know how long it takes to hit the ground. You know v₀ = 0, a = −9.8 m/s² (upward positive), and d is the height of the building (negative, since the ball moves downward). You don’t know final velocity and the problem doesn’t ask for it, so the missing variable is v. That points you to d = v₀t + ½at². With v₀ = 0, it simplifies to d = ½at², and you solve for t.
Projectile Motion: Horizontal vs. Vertical
Projectile problems split into two independent sets of equations, one for horizontal motion and one for vertical. Horizontally, there’s no acceleration (ignoring air resistance), so the horizontal velocity stays constant the entire flight. That means horizontal displacement is simply distance = velocity × time, and you don’t need the other kinematic equations at all for that direction.
Vertically, gravity provides a constant acceleration of 9.8 m/s² downward, so you use the standard four equations with a = −9.8 m/s². The vertical velocity changes by 9.8 m/s every second. Time is usually the variable that connects the two directions: solve for it in whichever direction gives you enough information, then carry it over to the other.
For example, a ball launched horizontally off a cliff has v₀ = 0 in the vertical direction. You’d use d = ½at² vertically to find the fall time, then use that time horizontally to find how far from the base of the cliff it lands.
When These Equations Don’t Apply
All four kinematic equations assume constant acceleration. If acceleration is changing, such as a car that speeds up, cruises, and then brakes, you can’t apply a single equation to the entire trip. Instead, break the motion into segments where acceleration stays constant within each one, solve each segment separately, and combine the results. The final velocity of one segment becomes the initial velocity of the next.
Situations with changing acceleration that can’t be broken into constant segments (like a bungee jumper whose acceleration varies continuously) require calculus-based methods rather than these algebraic equations.
Quick-Reference Decision Table
When you’ve identified the variable that’s missing from your problem, match it here:
- Don’t have or need d: v = v₀ + at
- Don’t have or need a: d = ½(v₀ + v)t
- Don’t have or need v: d = v₀t + ½at²
- Don’t have or need t: v² = v₀² + 2ad
The entire strategy fits into one sentence: find the variable your problem doesn’t care about, and use the equation that doesn’t contain it.