Kinematics Calculator: Solve for Any Variable

How to use the calculator above

Type any three of the five variables into the calculator above and it solves the constant-acceleration equations for whatever is missing. The five inputs match the symbols on the AP Physics 1 equation sheet: initial velocity $v_0$ (written $v_{x0}$ on the sheet), final velocity $v$, acceleration $a$, time $t$, and displacement $\Delta x$ (the sheet writes it as $x - x_0$).

Every kinematics equation connects four of these five quantities, so three known values are enough to pin down the remaining two. The calculator picks the equation that skips your unknown, substitutes your numbers, and reports each result with units. Use it to check homework, not to replace the setup: on the AP free-response section you still need to write the equation, show the substitution, and do the algebra yourself.

The three equations on the AP sheet

The AP Physics 1 equation sheet prints exactly three kinematics equations:

All three assume constant acceleration. Each one is missing exactly one variable: the first has no displacement, the second has no final velocity, and the third has no time.

A fourth equation you may have learned, $\Delta x = \frac{v_0 + v}{2} t$, is not printed on the sheet. It follows from the definition of average velocity: displacement equals average velocity times time, and with constant acceleration the average velocity is the average of $v_0$ and $v$. You are free to use it, just know where it comes from. For full derivations, see the kinematic equations guide.

Pick the equation by its missing variable

The fastest reliable method: list what you know, note what you want, and find the one variable that is neither. Then use the equation that does not contain that variable.

Say a problem gives $v_0$, $a$, and $\Delta x$ and asks for $v$. Time is neither given nor requested, so the no-time equation $v^2 = v_0^2 + 2a\Delta x$ finishes the job in one step. The calculator above runs this same selection logic, which makes it a quick way to confirm you picked the equation a grader would expect.

Sign conventions: the biggest source of wrong answers

Every variable in these equations except time is a vector component, so signs carry direction. Before substituting anything, pick a positive direction and apply it to all five variables at once.

• Ball thrown upward with up as positive: $v_0$ is positive and $a = -9.8 \ \text{m/s}^2$ for the entire flight, including at the very top.
• Car braking while moving in the positive direction: $v_0$ is positive and $a$ is negative. Deceleration just means the acceleration points opposite the velocity.
• A negative $\Delta x$ in your result means the object ended up on the negative side of its starting point. That is direction information, not an algebra mistake.

The calculator above uses exactly the signs you type, so enter $-9.8$ for a rising ball when up is positive. On free-response work, state your positive direction before the first substitution so every sign in your solution has a defined meaning.

When constant acceleration applies

These equations work only while the acceleration stays constant in magnitude and direction. That covers the classic Unit 1 situations: free fall near Earth's surface with $a = 9.8 \ \text{m/s}^2$ downward, cars speeding up or braking at a steady rate, and objects pushed by a constant net force. They do not cover a rocket burning off fuel or any interval where the acceleration changes value. For those, split the motion into constant-acceleration segments and chain them: the final velocity of one segment becomes the initial velocity of the next.

Projectiles are the special case you will meet most often: constant $9.8 \ \text{m/s}^2$ downward vertically and zero acceleration horizontally. Split the motion into axes and these same equations handle each axis separately, which is exactly what the projectile motion calculator automates. All of this sits in Topic 1.2 of AP Physics 1 Unit 1, which carries 10 to 15 percent of the multiple-choice section.

Common mistakes the calculator can catch

Solve by hand first, then compare with the calculator's output. When the two disagree, the cause is usually one of these:

• Unit mismatches: 72 km/h must become 20 m/s before it touches an equation. Convert everything to meters and seconds first.
• Dropping the $\frac{1}{2}$ in $\Delta x = v_0 t + \frac{1}{2} a t^2$, or forgetting to square the time.
• Entering $+9.8$ for gravity after declaring up as positive.
• Treating the top of a vertical arc as $v = 0$ in two-dimensional problems. Only the vertical component is zero there; the horizontal component never changes.
• Ignoring the second root when solving for time. Both roots of the quadratic can be physical, as the third worked example below shows.

If the setup itself feels shaky, the kinematic equations guide walks through choosing knowns and unknowns, and the projectile motion guide covers the two-dimensional version. Kinematics is the foundation the rest of the course builds on, so a fast start here is a real part of how hard AP Physics 1 actually is to manage.