Kinematics Calculator
Enter any three of the five constant-acceleration variables and leave the other two blank. The calculator identifies which kinematic equation applies, solves for the unknowns, and shows the formula and substitutions at each step. For background on the four equations and how to choose between them, see the Kinematic Equations lesson.
For constant-acceleration motion, five variables describe the motion: initial velocity \(v_0\), final velocity \(v\), acceleration \(a\), time \(t\), and displacement \(\Delta x\). Four equations relate them. If you know any three of the five, you can solve for the remaining two by picking the equation that includes what you know and omits the variable you don't.
The Four Kinematic Equations
| # | Equation | Omits |
|---|---|---|
| 1 | \(v = v_0 + at\) | \(\Delta x\) |
| 2 | \(\Delta x = v_0 t + \tfrac{1}{2} a t^2\) | \(v\) |
| 3 | \(v^2 = v_0^2 + 2a \Delta x\) | \(t\) |
| 4 | \(\Delta x = \tfrac{1}{2}(v_0 + v), t\) | \(a\) |
The "Omits" column tells you when each equation applies: pick the one that doesn't contain the variable you neither know nor need.
Worked Example
A ball is rolled at \(v_0 = 6\) m/s and decelerates at \(a = -0.5\) m/s². How far does it roll before stopping?
Knowns: \(v_0 = 6\), \(v = 0\), \(a = -0.5\). Unknown: \(\Delta x\). Missing: \(t\).
Use equation 3 (omits \(t\)):
\[\Delta x = \dfrac{v^2 - v_0^2}{2a} = \dfrac{0 - 36}{-1} = 36 \text{ m}\]
Tips for Using the Calculator
- "Starts from rest" means \(v_0 = 0\). "Comes to a stop" means \(v = 0\). Don't forget these implicit zeros.
- A dropped object near Earth's surface has \(v_0 = 0\) and \(a = g = 9.8\) m/s² (taking down as positive).
- An object slowing down has acceleration in the opposite direction of motion — its sign is negative relative to the direction of motion.
- Pick a sign convention (say, "right is positive") at the start of a problem and stick with it throughout.
Frequently Asked Questions
Why do I have to enter exactly three values?
The five variables are linked by four independent equations. Knowing three of them is enough to pin down the other two. Knowing fewer than three leaves multiple consistent motions; knowing more than three risks giving the calculator inconsistent inputs (no real motion satisfies all four constraints exactly).
Can I use this for a free-fall problem?
Yes. Free fall is constant-acceleration motion with \(a = 9.8\) m/s² (taking down as positive) or \(a = -9.8\) m/s² (taking up as positive). Enter that for \(a\) and provide whichever other two values the problem gives you. For a dedicated, simpler tool, see the Free Fall Calculator.
What if my problem has changing acceleration?
Then the kinematic equations don't apply directly — they're only valid when acceleration is constant. For piecewise-constant cases (one acceleration for the first phase, a different one for the second), solve each phase separately, using the final values of one phase as the initial values of the next.
What does a negative displacement mean?
Displacement is the vector difference between final and initial position. If you pick "right is positive" and an object ends up to the left of where it started, its displacement is negative. The signs of velocity, acceleration, and displacement all need to be consistent with the same convention.
Why are there two answers when I solve for time?
Equation 2 is quadratic in \(t\), so it can have two positive roots. Physically, both might be meaningful — for example, a ball thrown straight up passes through a given height twice, once on the way up and once on the way down. The calculator reports both solutions when this happens.