Kinematic Equations Calculator (SUVAT Solver)

Kinematic Equations Calculator

Kinematic Equations Formula

Formula

\[ v = u + at \\ s = ut + \frac{1}{2}at^2 \\ v^2 = u^2 + 2as \\ u = v - at \\ a = \frac{v - u}{t} \\ t = \frac{v - u}{a} \]

Where:

\(u\) — initial velocity (m/s)
\(v\) — final velocity (m/s)
\(a\) — constant acceleration (m/s²)
\(t\) — elapsed time (s)
\(s\) — displacement (m)

These are the SUVAT equations of motion with constant acceleration. Each equation links four of the five variables, so any one unknown can be found as soon as the other three are known.

Kinematic Equations – Calculation Example

Example: a motorcycle accelerates at \(4\ \text{m/s}^2\) for \(5\ \text{s}\) and reaches a final velocity of \(30\ \text{m/s}\). What was its initial velocity?

\[ u = v - at = 30 - 4 \times 5 = 30 - 20 = 10\ \text{m/s} \]

Result: the initial velocity was \(10\ \text{m/s}\) (about \(36\ \text{km/h}\)). You can reproduce this by entering 30 m/s, 4 m/s², 5 s and selecting Initial Velocity (u) in the solver.

How to Use the Kinematic Equations Calculator

Leave the field you want to solve for empty — the calculator hides it automatically when you select it.
Enter your known values in the Initial Velocity (u), Final Velocity (v), Acceleration (a), Time (t) and Displacement (s) fields.
Pick the matching unit for each value: m/s or km/h for velocities, s or min for time, m or km for displacement. Acceleration is always entered in m/s².
Open the Solve for dropdown and choose the variable you want: Initial Velocity (u), Final Velocity (v), Acceleration (a), Time (t) or Displacement (s).
Read the answer under Kinematic Results, shown in the same unit system you selected.

The solver converts km/h, minutes and kilometres to SI units internally, applies the correct rearranged SUVAT equation, and converts the result back to your chosen unit.

What Are the Kinematic Equations?

The kinematic equations — often called SUVAT equations after the five symbols \(s\), \(u\), \(v\), \(a\), \(t\) — describe the motion of an object moving along a straight line with constant acceleration. They are derived from the definitions of velocity and acceleration by calculus, but you do not need calculus to use them: each equation is simply a relationship between four of the five variables.

Which equation you need depends on what you know and what you want:

To find the final velocity from \(u\), \(a\) and \(t\): use \(v = u + at\).
To find displacement from \(u\), \(a\) and \(t\): use \(s = ut + \frac{1}{2}at^2\).
To link velocities and displacement without time: use \(v^2 = u^2 + 2as\).
To recover the initial velocity, acceleration or time, rearrange \(v = u + at\) into \(u = v - at\), \(a = \frac{v-u}{t}\) or \(t = \frac{v-u}{a}\).

This calculator applies the right rearrangement for you, which removes the most common source of mistakes in kinematics problems. It is useful for physics homework, vehicle dynamics, braking and acceleration estimates, and any straight-line motion where acceleration does not change.

If your problem involves two-dimensional motion, the same equations apply separately to the horizontal and vertical components — that is exactly how projectile motion is analyzed. For motion under gravity alone, see the free fall calculator, and for braking problems the stopping distance calculator is a natural companion.

Assumptions, Limitations and Practical Tips

Constant acceleration only. The SUVAT equations are invalid if acceleration changes during the motion (for example, air resistance growing with speed). For motion with varying acceleration you need calculus or numerical methods.
Straight-line motion. Displacement \(s\) is measured along the line of motion. In 2D or 3D problems, apply the equations to each coordinate axis separately.
Solving for time or acceleration uses velocities. The solver computes \(a = (v-u)/t\) and \(t = (v-u)/a\), so make sure the velocities you enter are consistent with the acceleration direction.
Watch for division by zero. Solving for \(t\) requires a non-zero acceleration; if \(a = 0\), the velocity never changes and time cannot be determined from velocities alone.
Signs matter. Velocities and acceleration can be negative. A negative result means the quantity points opposite to the direction you chose as positive — for instance, deceleration is simply negative \(a\).
Check consistency. If you enter all five values, verify they satisfy the equations (the default values do: \(10 + 4 \times 5 = 30\) and \(10 \times 5 + \frac{1}{2} \times 4 \times 25 = 100\)).

For everyday problems where acceleration is unknown and you only know distances and times, the average speed calculator may be the simpler tool.

Frequently Asked Questions

How do I choose the right SUVAT equation when solving by hand?

Write down the five variables \(s, u, v, a, t\) and mark the three you know plus the one you want. Each equation omits exactly one variable, so pick the equation that excludes the value you neither know nor need. If time is missing, use \(v^2 = u^2 + 2as\); if displacement is missing, use \(v = u + at\); if final velocity is missing, use \(s = ut + \frac{1}{2}at^2\).

How do I handle deceleration or braking problems?

Deceleration is just negative acceleration. Pick a positive direction (usually the direction of travel), then enter the velocity and acceleration with opposite signs. For example, a car braking from \(20\ \text{m/s}\) to a stop with \(a = -5\ \text{m/s}^2\) gives a stopping time of \(t = (0 - 20)/(-5) = 4\ \text{s}\). Keeping signs consistent is the single most important habit in kinematics.

Can I use these equations for free fall or projectile motion?

Yes. For free fall, set \(a = 9.81\ \text{m/s}^2\) downward (enter it as negative if you call upward positive) and \(u = 0\) for a dropped object. For projectiles, split the motion into two independent 1D problems: horizontal motion with \(a = 0\), and vertical motion with \(a = \pm 9.81\ \text{m/s}^2\). Solve each axis separately with the SUVAT equations, then combine the components.

How do I convert km/h to m/s and other common units?

Divide km/h by 3.6 to get m/s (so \(90\ \text{km/h} = 25\ \text{m/s}\)), or multiply m/s by 3.6 to go back. This works because \(1\ \text{km/h} = 1000\ \text{m}/3600\ \text{s}\). The calculator does this automatically, but when working by hand always convert everything to SI units (m, s, m/s, m/s²) before plugging numbers into the equations.

What is the difference between distance and displacement in these equations?

Displacement \(s\) is the net change in position, not the total path length. If you walk 5 m forward and 5 m back, the distance traveled is 10 m but the displacement is 0 m. The SUVAT equations always use displacement. This matters most when the object reverses direction — for example, a ball thrown up and caught at the same height has \(s = 0\) even though it traveled a considerable distance.

What are the most common mistakes in kinematics problems?

The top errors are: mixing units (entering km/h alongside seconds), forgetting that velocities and acceleration can be negative, using distance instead of displacement, and applying the equations when acceleration is not constant. Also watch for two-root situations: quadratic equations like \(s = ut + \frac{1}{2}at^2\) can give two mathematically valid times, and you must choose the physically sensible one (usually the positive, later value).

Why do I get two possible answers when solving for time or velocity?

Some rearranged SUVAT equations are quadratic, so two values satisfy the math. A ball thrown upward passes the same height twice — once going up, once coming down — so solving for time at that height yields two correct answers. Check both solutions against the physical situation: reject negative times, and if the problem asks for a specific phase of the motion (rising vs. falling), pick the root that matches.

Kinematic Equations Solver