Kinematic Equations Calculator (SUVAT)
- Free
- No sign-up
- Works on phones
- Shows the working
By Yasir Hafeez · Updated
This kinematic equations calculator solves constant-acceleration motion problems. Enter any three of the five SUVAT quantities (displacement \( s \), initial velocity \( u \), final velocity \( v \), acceleration \( a \), time \( t \)), leave the other two blank, and it finds them and tells you which equations it used.
It’s built for physics and calculus students working on free fall, braking, launches and other one-dimensional motion. It’s free and needs no sign-up.
How to use the kinematic equations calculator
- Pick a positive direction (for example, down for a dropped object or forward for a car) and keep to it.
- Enter exactly three known values in metres, metres per second and seconds. Use a negative number for anything pointing the other way, such as braking acceleration.
- Leave the two unknowns blank.
- Press Solve. The table shows all five values, and the note lists the equations used.
The five kinematic equations
For motion in a straight line with constant acceleration:
$$v = u + at$$
$$s = ut + \tfrac12at^2$$
$$v^2 = u^2 + 2as$$
$$s = \tfrac12(u + v)t$$
$$s = vt - \tfrac12at^2$$
Each equation leaves out one of the five quantities, so you choose the one missing the quantity you neither know nor want.
| Equation | Doesn’t contain |
|---|---|
| \( v = u + at \) | \( s \) |
| \( s = ut + \frac12at^2 \) | \( v \) |
| \( v^2 = u^2 + 2as \) | \( t \) |
| \( s = \frac12(u + v)t \) | \( a \) |
| \( s = vt - \frac12at^2 \) | \( u \) |
Worked example: free fall
A stone is dropped from rest. How far does it fall in 3 seconds, and how fast is it going? Take down as positive, so \( u = 0 \), \( a = 9.81 \) m/s², \( t = 3 \) s.
$$v = 0 + 9.81 \cdot 3 = 29.43 \text{ m/s}$$
$$s = 0 + \tfrac12(9.81)(3^2) = 44.145 \text{ m}$$
These are the values in the calculator above.
Worked example: braking distance
A car brakes from some speed to a stop in 50 m with acceleration \( -4 \) m/s². What was its initial speed, and how long did braking take? Known: \( s = 50 \), \( v = 0 \), \( a = -4 \).
Initial speed. \( v^2 = u^2 + 2as \) gives \( 0 = u^2 - 400 \), so \( u = 20 \) m/s (72 km/h).
Time. \( t = \frac{v - u}{a} = \frac{0 - 20}{-4} = 5 \) s.
Worked example: finding acceleration
A cyclist speeds up from 5 m/s to 15 m/s in 4 s. Then \( a = \frac{15 - 5}{4} = 2.5 \) m/s² and \( s = \frac12(5 + 15)(4) = 40 \) m.
Tips and common mistakes
- Constant acceleration only. If the acceleration changes, these equations don’t apply. Use \( v(t) = \int a(t)\,dt \) instead (see the position, velocity and acceleration calculator).
- Sign errors. Gravity is \( -9.81 \) if up is positive and \( +9.81 \) if down is positive.
- Mixed units. Convert km/h to m/s by dividing by 3.6 before you start.
- Two square roots. \( v^2 = u^2 + 2as \) gives \( \pm \); choose the sign that matches the direction of motion. The calculator picks the one with non-negative time and says so.
Where the equations come from
They are calculus in disguise. Constant acceleration integrates to \( v(t) = u + at \), and integrating again gives \( s(t) = ut + \frac12at^2 \). Eliminate t between them and you get \( v^2 = u^2 + 2as \). The kinematic equations guide walks through each derivation, and the fundamental theorem of calculus explains why integrating velocity gives displacement.
Related guides
- Kinematic Equations: The 5 SUVAT Formulas
- Projectile Motion
- Related Rates
- Fundamental Theorem of Calculus
Further reading
- 3.4 Motion with Constant Acceleration (OpenStax University Physics Volume 1) — the kinematic equations with worked problems.
- Equations of motion (Wikipedia) — the SUVAT equations and their calculus derivation.
FAQ
What are the kinematic equations?
\( v = u + at \), \( s = ut + \frac12at^2 \), \( v^2 = u^2 + 2as \), \( s = \frac12(u + v)t \) and \( s = vt - \frac12at^2 \), valid for constant acceleration.
How many values do I need?
Three. With any three of s, u, v, a and t, the other two are determined.
What does SUVAT stand for?
The five variables: s for displacement, u for initial velocity, v for final velocity, a for acceleration and t for time.
Can I use it for objects thrown upward?
Yes. Take up as positive, enter \( a = -9.81 \), and the launch speed as u. At the highest point \( v = 0 \).