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. Constant acceleration guarantees that average velocity equals the endpoint mean $\frac{v_0+v}{2}$; that equality is not generally valid for changing acceleration, although it can hold in particular cases. 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 $t$. Each of the four equations listed here omits one of these quantities. They are useful forms, not an exhaustive list of every algebraically equivalent relation.

This gives a useful first 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 $v_{x0}$ appears in all four listed forms. If it is the quantity you neither know nor need, eliminate it with $v_{x0}=v_x-a_xt$, giving $\Delta x=v_xt-\frac{1}{2}a_xt^2$. This equivalent relation avoids solving for an unnecessary initial velocity. 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 from rest with negligible air resistance near Earth 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 an unnecessary unknown and pick an equation that omits it, using the table above or the derived form without initial velocity.
5. Solve algebraically first, then substitute numbers with units. Rearranging symbols before plugging in makes sign errors easier to catch and keeps the target visible.

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

Fix a positive direction before touching any numbers.

• With up as positive, the acceleration in near-Earth free fall with negligible air resistance is $-9.8 \, \mathrm{m/s^2}$ for the entire flight, on the way up, at the top, and on the way down.
• This guide and its calculator use $g=9.8 \, \mathrm{m/s^2}$. The AP Physics 1 CED uses approximately $10 \, \mathrm{m/s^2}$ for numerical questions and explicitly permits correct use of $9.8$ or $9.81 \, \mathrm{m/s^2}$; its information table prints $9.8$. Follow the value specified in your problem and keep it consistent.
• 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. Do not apply one constant acceleration across an interval where it changes. If the motion has separate constant-acceleration phases, solve each phase and pass its final state to the next. More general motion may require graph areas, energy methods, or calculus, as with a mass in simple harmonic motion.

In two dimensions they still work, one axis at a time. Ideal projectile motion near Earth, neglecting air resistance, is an 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.

They are the constant-acceleration special case of integrating $a(t)$ twice, which is how AP Physics C derives them: rectilinear motion.

Check the signs, then try a new problem

Take right as positive. A cart has $v_0=12\ \mathrm{m/s}$ and constant $a=-3.0\ \mathrm{m/s^2}$ for $2.0\ \mathrm{s}$. Before calculating, predict whether its final velocity is positive and whether it has reversed. Find its displacement.

Check: $v=12-3.0(2.0)=6.0\ \mathrm{m/s}$, so it is still moving right. Its displacement is $\Delta x=12(2.0)+\frac{1}{2}(-3.0)(2.0)^2=18\ \mathrm{m}$.

In the kinematics calculator, choose dx, enter v0=12, a=-3, and t=2, and clear the v field. The default fields contain a different example; do not leave an unrelated value filled in. When solving a quadratic for time, the calculator prefers the smallest positive time root, using zero only when no positive root is available, so a readout does not replace checking which event the question asks about.

Continue to kinematics practice: begin with “Choose the equation before you calculate,” then try “Reading a velocity-time graph,” and finish with “Reaction time plus braking.” Keep the solutions closed until you have identified the interval, axis, and equation. The progression moves from equation choice to graph interpretation to a two-phase calculation.

For the constant-acceleration assumptions and derivations, see OpenStax, Motion with Constant Acceleration.