SUVAT Calculator (Kinematic Equations)
Any three of s, u, v, a, t in — the other two out, with both roots and every substitution.
Motion graphs
Displacement against time
Velocity against time
Values over time
How it was worked out
About the SUVAT Calculator (Kinematic Equations)
Solve any constant-acceleration (SUVAT) problem: type any three of displacement s, initial velocity u, final velocity v, acceleration a and time t, and the calculator finds the other two. For each unknown it picks the equation that does not contain the other unknown — exactly the method taught in physics classes — and writes out every substitution.
When the time comes from a quadratic (for example s, u and a known), both roots are shown and paired with their own velocity: a ball thrown upwards passes the same height once on the way up and once on the way down. Roots with a negative time are kept but marked as rejected. Values can be in m, km, ft or miles, m/s, km/h or mph, m/s² or g, and seconds, minutes or hours, and a vertical-motion preset fills in a = −g or +g for Earth, the Moon or the planets.
Displacement–time and velocity–time graphs and a table of values (downloadable as CSV) show the whole motion.
How to use it
- Choose a positive direction — up, or forwards — and keep to it: anything pointing the other way is negative (use the ± button on a phone).
- Type three of the five values and leave the two you want to find blank. For free fall, pick the gravity and press Up is positive (a = −g) or Down is positive (a = +g).
- Read the answer under each blank box and in the solution cards. If there are two solutions, both are shown with the direction of motion at that moment.
- Check the equations and substitutions in How it was worked out, look at the graphs, and copy the result or download the table as CSV.
Examples
s = 15 m, u = 20 m/s, a = −9.81 m/s² (up positive)
t = 0.9907 s with v = +10.28 m/s (rising), or t = 3.087 s with v = −10.28 m/s (falling)
s = 20 m, u = 0, a = 9.81 m/s² (down positive)
t = 2.019 s, v = 19.81 m/s (the root t = −2.019 s is rejected)
u = 36 km/h, v = 108 km/h, t = 30 s
a = 0.6667 m/s², s = 600 m
u = 5 m/s, v = 11 m/s, a = 2 m/s²
s = (v² − u²) ÷ 2a = 24 m, t = (v − u) ÷ a = 3 s
s = 24 m, v = 11 m/s, a = 2 m/s²
u = 5 m/s and t = 3 s, or u = −5 m/s and t = 8 s
s = 200 m, u = 10 m/s, a = 2 m/s²
t² + 10t − 200 = 0 → t = 10 s, v = 30 m/s
Common uses
- GCSE, A-level, IB, AP and first-year university kinematics problems, with the working to check against.
- Vertical throws and drops: maximum height, time in the air and impact speed (without air resistance).
- Braking and acceleration problems for cars, trains and aircraft.
- Checking which equation to use, and seeing why a quadratic gives two times.
The five SUVAT equations
For motion in a straight line with constant acceleration (s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time):
v = u + a t— no ss = u t + ½ a t²— no vs = v t − ½ a t²— no uv² = u² + 2 a s— no ts = ½ (u + v) t— no a
Each equation leaves out one variable. With three known values and two unknowns, find each unknown with the equation that leaves out the other unknown.
In OpenStax College Physics 2e the first, second and fourth are Eqs. 2.52–2.54 (with x − x₀ = s and v₀ = u). The fifth combines Eqs. 2.50 and 2.51 (x = x₀ + v̄t with v̄ = (v₀ + v) ÷ 2), and the third follows from the second by substituting u = v − a t.
Why there can be two answers
When s, u and a are known, s = u t + ½ a t² is a quadratic in t, so there are two roots: t = (−u ± √(u² + 2as)) ÷ a. A ball thrown upwards at 20 m/s is 15 m up after 0.99 s (still rising) and again after 3.09 s (falling back). A negative root describes a moment before t = 0 and is rejected. If u² + 2as is negative, the object never reaches that displacement. With s, v and a known, the two roots can even be two different motions with different starting velocities.
Signs and the value of g
Choose one direction as positive. If up is positive, gravity gives a = −g; if down is positive, a = +g. Near Earth’s surface g varies from about 9.78 to 9.83 m/s² with latitude and height; textbooks round it to 9.81 or 9.8 m/s², and the standard value is g₀ = 9.80665 m/s². The preset also offers the Moon (1.62 m/s²) and the planets, from NASA JPL data. These results ignore air resistance — for falls with drag, use the free fall calculator.
Sources
- OpenStax, College Physics 2e, §2.5 Motion Equations for Constant Acceleration in One Dimension (Eqs. 2.50–2.54).
- OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Eqs. 3.10–3.14, Example 3.11) and §3.5 Free Fall (sign convention, g from 9.78 to 9.83 m/s²).
- BIPM, 3rd CGPM (1901), declaration on the conventional value of gₙ (980.665 cm/s²); NASA JPL, Planetary Physical Parameters (gravity of other bodies).
Limitations
- The SUVAT equations need a constant acceleration and motion along one straight line. For curved paths, split the motion into components (see the projectile motion calculator).
- Air resistance is not included.
- Times must be positive when entered; a negative time root is shown but rejected.
- Speeds close to the speed of light need relativity; faster-than-light values are refused.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
What does SUVAT stand for?
The five quantities of constant-acceleration motion: s displacement, u initial velocity, v final velocity, a acceleration and t time. Knowing any three lets you find the other two.
Which SUVAT equation should I use?
Each equation leaves one variable out. Find the variable you neither know nor need, and use the equation without it. To find v when t is not given, use v² = u² + 2as; to find s when a is not given, use s = ½(u + v)t.
Why do I get two values for time?
Because s = ut + ½at² is a quadratic in t. For a ball thrown up at 20 m/s, s = 15 m happens at t = 0.99 s on the way up and t = 3.09 s on the way down. If one root is negative, it is a time before the motion started and is rejected.
Should g be positive or negative?
It depends on which way you call positive. With up as positive, a = −9.81 m/s² (gravity pulls down); with down as positive, a = +9.81 m/s². Keep every other value consistent with the same choice.
Can I use SUVAT when the acceleration changes?
No — the equations assume a constant acceleration. If it changes in stages, solve each stage separately and use the final velocity of one stage as the initial velocity of the next.
What is the difference between displacement and distance?
Displacement is the change in position, with a sign for direction; distance is the length of path travelled. A ball thrown up and caught at the same height has a displacement of 0, even though it travelled up and down.