Kinematics Calculator
Enter any three of displacement, initial velocity, final velocity, acceleration and time, and the calculator finds the other two. It picks the right constant-acceleration (SUVAT) equation for your three knowns, shows every rearrangement and substitution, and draws the velocity–time graph. Runs entirely in your browser — free, no sign-up.
Enter any three of the five quantities and leave the other two blank — the blanks are calculated. Use a minus sign for velocity or acceleration in the opposite direction.
Calculated
Calculated
Results
160 m
Displacement s
2 m/s²
Acceleration a
20 m/s
Average velocity (s ÷ t)
The working
- Known: u = 12 m/s, v = 28 m/s, t = 8 s. Unknown: s, a.
- Find a from v = u + at, rearranged as a = (v − u) / t.
- a = (28 − 12) / 8 = 2 m/s²
- Find s from s = ½(u + v)t, rearranged as s = (u + v) × t / 2.
- s = (12 + 28) × 8 / 2 = 160 m
Velocity–time graph
The slope of the line is the acceleration a. The shaded area between the line and the time axis is the displacement s; area below the axis counts as negative.
Assumes constant acceleration along a straight line. Where the acceleration changes, or the motion is in two dimensions, these equations do not apply.
How to Use the Kinematics Calculator
- 1. Decide on a direction. Pick one direction as positive. Displacement, velocity and acceleration are signed along it, so a ball thrown upward has a positive u and a negative a (gravity pulls down).
- 2. Fill in exactly three boxes. Choose a unit for each — metres or feet, m/s or mph, seconds or minutes — and leave the other two blank. The blanks turn into answers.
- 3. Read the answers and the working. The calculator converts to SI units, names the equation it used for each unknown, and shows the numbers going in.
- 4. Check the graph. The line runs from u to v over the time t. Its slope is the acceleration and the shaded area under it is the displacement.
The Five Kinematic Equations
Constant-acceleration motion in a straight line is described by five variables — s, u, v, a and t — and five equations. Each equation leaves out exactly one variable, which is how you choose between them: find the quantity you do not know and do not care about, and use the equation without it.
| Equation | Leaves out | Use it when you know |
|---|---|---|
| v = u + at | s | u, a, t (to find v) or u, v, a (to find t) |
| s = ut + ½at² | v | u, a, t (to find s) or s, u, a (to find t) |
| s = vt − ½at² | u | v, a, t (to find s) or s, v, a (to find t) |
| s = ½(u + v)t | a | u, v, t (to find s) or s, u, v (to find t) |
| v² = u² + 2as | t | u, a, s (to find v) or u, v, a (to find s) |
Only two of the five are independent: the others follow by eliminating a variable. Source: OpenStax University Physics, Volume 1, section 3.4.
Any three of the five variables fix the other two, which gives ten possible problems. The calculator handles all ten, including the ones that need the quadratic formula.
Worked Examples
A car speeds up. A car goes from 12 m/s to 28 m/s in 8 s. Known: u, v, t. From v = u + at, a = (28 − 12) / 8 = 2 m/s². From s = ½(u + v)t, s = ½ × (12 + 28) × 8 = 160 m. The average velocity is 160 / 8 = 20 m/s, exactly halfway between u and v, as it must be when the acceleration is constant.
A car brakes. A car at 25 m/s (90 km/h) stops in 60 m. Known: s, u, v with v = 0. There is no time, so use v² = u² + 2as: a = (0 − 25²) / (2 × 60) = −5.208 m/s². The minus sign says the acceleration points opposite to the motion. Then t = 2s / (u + v) = 2 × 60 / 25 = 4.8 s.
Two answers. A ball is thrown straight up at 20 m/s and you ask when it is 15 m above the hand, taking a = −9.80665 m/s². Known: s, u, a. The equation s = ut + ½at² is a quadratic in t, and both roots are valid: t = 0.9906 s on the way up and t = 3.088 s on the way down. The final velocities are +10.29 m/s and −10.29 m/s — the same speed, opposite directions. The calculator lists both solutions and lets you switch between them.
Reading the Velocity–Time Graph
With constant acceleration the velocity–time graph is a straight line. Its slope is a, and the signed area between the line and the time axis is s. Where the line crosses zero the object turns around; area below the axis counts as negative displacement, so the net displacement can be smaller than the distance actually travelled.
Assumptions and When There Is No Answer
- Constant acceleration, one dimension. If the acceleration changes (a car that eases off the throttle, a falling object with significant air drag), these equations only describe each short stretch. For motion in two dimensions, use the Projectile Motion Calculator.
- Free fall. The calculator has no built-in g; type a = −9.80665 m/s² (standard gravity, an exact definition) or choose the unit g. Real values range from about 9.78 m/s² at the equator to 9.83 m/s² at the poles.
- Time moves forward. A negative time is rejected, and a mathematical root that needs negative time is left out. If no real solution exists — a ball that never reaches 100 m, or speeds that cannot connect at that acceleration — the calculator says so rather than returning nonsense.
- Undetermined cases. Three values sometimes do not pin the motion down, such as a = 0 with u = v, where any time fits. The calculator asks you to change one value.
Frequently Asked Questions
- What does SUVAT stand for?
- S is displacement, U initial velocity, V final velocity, A acceleration and T time. The SUVAT equations are the five relationships between these five quantities that hold when acceleration is constant.
- Which kinematic equation should I use?
- List what you know and what you want, then pick the equation that contains all four of those quantities and leaves out the fifth. If time is neither given nor wanted, v² = u² + 2as is the one; if displacement is irrelevant, use v = u + at.
- Why do I sometimes get two answers?
- When you give s, u and a (or s, v and a), time appears squared in the equation, so it is a quadratic with two roots. A thrown ball passes a given height twice, once going up and once coming down, and both are correct. The calculator drops a root only when it is negative.
- Can acceleration or velocity be negative?
- Yes. The sign only tells you the direction relative to the one you chose as positive. Negative acceleration with positive velocity means slowing down; negative acceleration with negative velocity means speeding up in the negative direction.
- Is displacement the same as distance?
- No. Displacement s is a signed straight-line change in position, while distance is the total path length and is never negative. They match only if the object never reverses direction.
- What value of g should I use for free fall?
- Standard gravity is exactly 9.80665 m/s², and 9.8 or 9.81 are common roundings. Use the same value your course uses, and enter it as a negative number when upward is positive.