Kinematic Equations: The 4 Formulas and When to Use Each

The three equations on the AP sheet (plus the fourth)

The kinematic equations relate displacement, velocity, acceleration, and time for motion with constant acceleration, and the AP Physics 1 equation sheet prints exactly three of them:

Here $x_0$ and $x$ are initial and final position, $v_{x0}$ and $v_x$ are initial and final velocity, $a_x$ is the constant acceleration, and $t$ is time. These exact forms appear on the AP Physics 1 formula sheet.

The classic fourth equation is not printed on the sheet:

It says displacement equals average velocity times time; the average velocity equals the simple mean $\frac{v_0 + v}{2}$ only when acceleration is constant, which is why the equation needs that condition. You are free to use it on the exam, but you have to remember it yourself or derive it by averaging the initial and final velocities.

The trick: each equation is missing one variable

Constant-acceleration problems involve five quantities: displacement $(x - x_0)$, initial velocity $v_{x0}$, final velocity $v_x$, acceleration $a_x$, and time $t$. Every kinematic equation contains four of the five, so each equation is missing exactly one variable.

That is the whole selection method. A typical problem hands you three quantities and asks for a fourth. The fifth variable, the one you neither know nor need, tells you which equation to grab: pick the one that does not contain it.

Say a car brakes from 24 m/s to rest with a known acceleration, and the question asks for stopping distance. You know $v_{x0}$, $v_x$, and $a_x$; you want $x - x_0$. Time never appears anywhere, so use the equation with no $t$ in it: $v_x^2 = v_{x0}^2 + 2 a_x (x - x_0)$. No guessing, and no solving the same problem three different ways to see which works.

Which equation to use: keyed by the missing variable

Read this table by asking one question: which variable do I neither know nor need?

Two things to notice. Initial velocity $v_{x0}$ shows up in all four equations, so no single equation skips it; if $v_{x0}$ is unknown, solve for it first or combine two equations. And the acceleration-free equation is the one the sheet leaves out, so when a problem gives you two velocities and a time, either recall $\Delta x = \frac{v_0 + v}{2} t$ or find $a_x$ first from $v_x = v_{x0} + a_x t$ and finish with a sheet equation.

How to solve kinematics problems in five steps

1. Choose an axis and a positive direction. Write it down. Every sign in the problem follows from this choice.
2. List the knowns with signs and units. Translate the words: starts from rest means $v_{x0} = 0$, comes to a stop means $v_x = 0$, dropped means $v_{x0} = 0$ with acceleration of magnitude $9.8 \, \mathrm{m/s^2}$ pointing down.
3. Identify the target variable. Underline what the question actually asks for, including which instant or interval it refers to.
4. Spot the missing fifth variable and pick the equation that omits it, using the table above.
5. Solve algebraically first, then substitute numbers with units. Rearranging symbols before plugging in makes sign errors easier to catch and mirrors how AP free-response solutions are laid out.

After you solve, plug the same numbers into the kinematics calculator to check your arithmetic before moving to the next problem.

Sign conventions and the mistakes that cost points

Most kinematics errors are sign errors, so fix a positive direction before touching any numbers.

• With up as positive, the acceleration of anything in free fall is $-9.8 \, \mathrm{m/s^2}$ for the entire flight, on the way up, at the top, and on the way down.
• Use $g = 9.8 \, \mathrm{m/s^2}$, the value printed on the AP equation sheet, not 9.81.
• At the peak of a throw, velocity is zero but acceleration is still $9.8 \, \mathrm{m/s^2}$ downward. Setting acceleration to zero at the top is a classic trap.
• Deceleration is not automatically negative. Acceleration is negative when it points opposite your chosen positive direction, so an object speeding up while moving in the negative direction also has negative acceleration.
• Keep units consistent: convert km/h to m/s before substituting, and carry units through every line so a wrong equation announces itself.

Where these equations stop working (and where they show up next)

The kinematic equations assume constant acceleration. The moment acceleration changes with time or position, a car easing off the brakes, a mass on a spring, anything in simple harmonic motion, they no longer apply, and you switch to energy methods or (in Physics C) calculus.

In two dimensions they still work, one axis at a time. Projectile motion is the big application: $a_x = 0$ horizontally and $a_y = -9.8 \, \mathrm{m/s^2}$ vertically with up positive, and time is the shared variable linking the two axes. That method gets a full walkthrough in the projectile motion guide, and you can test launch angles yourself in the projectile launcher interactive.

The same equations reappear in Unit 5 with angular variables swapped in; see rotational kinematics. For the rest of the unit, including vectors and relative motion, start at the Unit 1 kinematics overview. Kinematics is the foundation the whole course builds on, so getting it automatic early is a big part of how hard AP Physics 1 actually is in practice.