Kinematic Equations Calculator

Solve any kinematics problem - enter three known values to find the other two.

📖 What are Kinematic Equations?

Kinematic equations, often called the SUVAT equations, describe the motion of an object undergoing uniform (constant) acceleration in a straight line. They relate five physical quantities - displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t) - through four equations. Given any three of these five values, you can use the equations to find the other two.

The equations were developed from the fundamental definitions of velocity and acceleration and form the backbone of classical mechanics. They are studied in every high school and university physics course worldwide, and they underpin the analysis of projectile motion, vehicle braking distances, rocket trajectories, free fall, and countless other real-world scenarios.

The key constraint is the assumption of constant acceleration. This means the acceleration does not change during the time interval being analysed. Free fall near Earth's surface is the most common real-world example - gravity provides a constant downward acceleration of approximately 9.81 m/s², making the SUVAT equations directly applicable when air resistance is negligible.

When acceleration is not constant - for example, a rocket burning fuel at varying rates, or a car with engine throttle mapped to a non-linear force curve - the SUVAT equations are no longer valid. In those cases, differential equations and calculus-based kinematics or numerical simulation are required.

Understanding kinematics is the essential first step before studying dynamics (which introduces forces via Newton's laws). You learn to describe motion before you learn to explain its causes. The two subjects together form the foundation of all mechanics.

📐 Formula

The four SUVAT equations:

1. v = u + at - links v, u, a, t (no displacement)

2. s = ut + ½at² - links s, u, a, t (no final velocity)

3. v² = u² + 2as - links v, u, a, s (no time)

4. s = ½(u + v)t - links s, u, v, t (no acceleration)

Variables:

- s = Displacement in metres (m) - distance from start to end, with direction

- u = Initial velocity in m/s - velocity at the start of the time interval

- v = Final velocity in m/s - velocity at the end of the time interval

- a = Acceleration in m/s² - constant rate of change of velocity

- t = Time in seconds (s) - duration of the motion

The four SUVAT equations:

1. v = u + at - links v, u, a, t (no displacement) 2. s = ut + ½at² - links s, u, a, t (no final velocity) 3. v² = u² + 2as - links v, u, a, s (no time) 4. s = ½(u + v)t - links s, u, v, t (no acceleration)

Variables: - s = Displacement in metres (m) - distance from start to end, with direction - u = Initial velocity in m/s - velocity at the start of the time interval - v = Final velocity in m/s - velocity at the end of the time interval - a = Acceleration in m/s² - constant rate of change of velocity - t = Time in seconds (s) - duration of the motion

📖 How to Use This Calculator

Identify three known values from your problem (s, u, v, a, or t).

Enter those three values in the corresponding fields. Leave the other two blank.

Click Calculate to see all five values solved.

If you see a dash (—) for a value, it means the combination of knowns you entered does not uniquely determine that variable with the current implementation - try providing a different combination of three knowns.

💡 Example Calculations

Example 1 - Braking Distance

A car travelling at 30 m/s (108 km/h) brakes with a deceleration of 8 m/s² until it stops (v = 0).

- Known: u = 30 m/s, v = 0 m/s, a = −8 m/s²

- Using v² = u² + 2as: 0 = 900 + 2(−8)s → s = 900 / 16 = 56.25 m

- Time: t = (v − u) / a = (0 − 30) / (−8) = 3.75 s

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Example 2 - Free Fall

A stone is dropped from rest (u = 0) from a cliff. After 4 seconds, how far has it fallen and how fast is it moving?

- Known: u = 0 m/s, a = 9.81 m/s² (downward), t = 4 s

- Displacement: s = ut + ½at² = 0 + 0.5 × 9.81 × 16 = 78.48 m

- Final velocity: v = u + at = 0 + 9.81 × 4 = 39.24 m/s

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Example 3 - Acceleration of a Sprinter

A sprinter reaches 10 m/s from rest over a distance of 20 m. What was the average acceleration?

- Known: u = 0 m/s, v = 10 m/s, s = 20 m

- Using v² = u² + 2as: 100 = 0 + 2 × a × 20 → a = 100 / 40 = 2.5 m/s²

- Time: t = (v − u) / a = 10 / 2.5 = 4 s

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Frequently Asked Questions

What are the SUVAT equations?+

SUVAT is an acronym for the five kinematic variables in uniform acceleration problems: S (displacement), U (initial velocity), V (final velocity), A (acceleration), and T (time). The four SUVAT equations are: v = u + at; s = ut + ½at²; v² = u² + 2as; s = ½(u + v)t. Each equation relates four of the five variables, so given any three values you can always find the remaining two using two different equations.

What is uniform acceleration?+

Uniform acceleration means the acceleration is constant - it does not change with time. This is a simplifying assumption that makes the SUVAT equations valid. Common real-world examples include free fall near Earth's surface (ignoring air resistance), a car accelerating at constant throttle on a straight road, or a ball rolling down a frictionless inclined plane. When acceleration varies, more advanced methods are required.

Can I use negative values for velocity or acceleration?+

Yes, and you often must. Velocity and acceleration are vectors, so their sign indicates direction. If you define rightward or upward as positive, then leftward or downward values are negative. For example, a ball thrown upward has initial velocity u = +20 m/s and acceleration a = −9.81 m/s² (gravity acts downward). This calculator accepts negative values for all fields except where physically impossible.

What happens when I enter more or fewer than three values?+

You must enter exactly three of the five values (u, v, a, t, s) for the calculator to solve for the remaining two. If you enter only two values, there are infinitely many solutions. If you enter all five, the calculator will check consistency. If fewer than three are provided, the calculator will prompt you to add more values.

How do I solve a problem where an object is thrown upward?+

Define upward as positive. Set initial velocity u = the launch speed (positive), acceleration a = −9.81 m/s² (gravity, downward = negative). At maximum height, final velocity v = 0. Enter u, a, and v = 0 to find the time to peak and the maximum height (displacement s). To find time and position when it returns to the ground, set s = 0 and solve for t - you will get two solutions (t = 0 at launch, and t = total flight time).

What are the 5 kinematic equations (SUVAT)?+

The 5 SUVAT equations for uniform acceleration in a straight line: (1) v = u + at. (2) s = ut + (1/2)at^2. (3) v^2 = u^2 + 2as. (4) s = ((u+v)/2) x t. (5) s = vt - (1/2)at^2. Where s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. Each equation links 4 of the 5 variables, so knowing any 3 allows you to find the remaining 2.

How do you solve a free fall problem with kinematics?+

For free fall (no air resistance), use g = 9.8 m/s^2 (downward). Set downward as positive. Example: an object is dropped from rest from 45 m. Find time to hit ground: s = ut + (1/2)at^2. s = 45 m, u = 0, a = 9.8 m/s^2. 45 = 0 + (1/2)(9.8)t^2. t^2 = 45/4.9 = 9.18. t = 3.03 seconds. Final velocity: v = u + at = 0 + 9.8 x 3.03 = 29.7 m/s.

Can kinematic equations be used for non-constant acceleration?+

No. The four standard kinematic equations (SUVAT equations) only apply when acceleration is constant. For variable acceleration, you need calculus: velocity is the integral of acceleration with respect to time, and displacement is the integral of velocity. Common examples of non-constant acceleration include a car applying brakes (deceleration varies), a falling object experiencing air resistance (drag force changes with speed), and a rocket burning fuel (thrust-to-weight ratio changes). For these cases, numerical integration or differential equations are required.

Which kinematic equation should I use when time is unknown?+

Use v squared = u squared + 2as (the time-independent equation) when you know initial velocity, final velocity, and acceleration, but not time. For example, to find stopping distance: v = 0, u = initial speed, a = deceleration. Solving: s = (v squared minus u squared) / (2a). This is the basis for braking distance calculations in road safety.

🔗 Related Calculators

📌 Quick Tips

💡The SUVAT equations only apply to motion with uniform (constant) acceleration. If acceleration varies with time, you need calculus-based kinematics or numerical integration.

💡When an object is in free fall near Earth's surface, acceleration a = 9.81 m/s² (downward). Set the downward direction as positive and acceleration as +9.81 to simplify sign handling.

💡Check your answer by substituting all five values back into a different SUVAT equation. If both equations are satisfied, your solution is correct.