Kinematics Solver — Solve SUVAT Equations for Motion Problems

The SUVAT Variables

The Five SUVAT Equations

How to Use the Kinematics Solver

1. Open the kinematics solver.
2. Identify which three SUVAT variables you know from the problem. Enter their values in the appropriate fields.
3. Leave the two unknown fields blank (or enter '?').
4. Click Solve. The tool selects the appropriate SUVAT equation(s), substitutes your values, and returns both unknown variables with step-by-step working.
5. Check the sign conventions: positive and negative directions must be consistent throughout the calculation. If an object falls downward and you define downward as negative, then a = −9.81 m/s² and any downward displacement is negative.

Sign Conventions — the Most Common Source of Error

Kinematics calculations fail most often because of inconsistent sign conventions. Before starting any problem, define your positive direction. Once chosen, it must remain consistent for all variables in the problem.

Example: a ball is thrown upward with initial speed 15 m/s. Choose upward as positive. Then:

• u = +15 m/s (initial velocity is upward = positive)
• a = −9.81 m/s² (gravity acts downward = negative)
• At the peak: v = 0 m/s
• If it lands below the launch point: s will be negative

If you switch the positive direction mid-problem, all your signs must change. The kinematics solver enforces sign consistency — enter a negative value as a negative number when the direction is opposite to your chosen positive direction.

Real-World Applications

Worked Example: Car Braking to a Stop

A car travelling at 30 m/s applies its brakes and decelerates at 6 m/s². How far does it travel before stopping?

Known: u = 30 m/s, v = 0 m/s (stops), a = −6 m/s² (deceleration, so negative). Unknown: s.

The equation that omits t is: v² = u² + 2as

Substituting: 0² = 30² + 2 × (−6) × s

0 = 900 − 12s → 12s = 900 → s = 75 m

The car needs 75 metres to stop. Enter these values into the solver to verify and see the step-by-step working formatted clearly.

Worked Example: Object in Free Fall

A stone is dropped from a bridge 45 m above a river. How long does it take to hit the water, and what is its speed at impact? (g = 9.81 m/s², downward positive.)

Known: u = 0 m/s (dropped, not thrown), a = +9.81 m/s², s = +45 m. Unknown: t and v.

For t, use s = ut + ½at²: 45 = 0 + ½ × 9.81 × t² → t² = 90/9.81 ≈ 9.174 → t ≈ 3.03 s

For v, use v = u + at: v = 0 + 9.81 × 3.03 ≈ 29.7 m/s

The stone hits after ~3.03 seconds at ~29.7 m/s (about 107 km/h).

When SUVAT Equations Do Not Apply

SUVAT equations assume constant acceleration. If the acceleration changes during the motion — for example, a rocket burning fuel (decreasing mass → changing acceleration), or air resistance that increases with speed — SUVAT does not apply directly. In these cases you need calculus (integration of the equation of motion) or numerical methods.

For the majority of introductory physics problems — projectile motion without air resistance, objects sliding on frictionless surfaces, constant engine thrust — SUVAT is the correct tool. The kinematics solver explicitly assumes constant acceleration; if your problem involves changing acceleration, you will need a different approach.

Projectile Motion — Horizontal and Vertical Independence

For projectile motion (an object launched at an angle), the horizontal and vertical components of motion are independent and must be treated separately.

Vertical: Apply SUVAT with u = u×sin(θ), a = −9.81 m/s², and the vertical constraints (v = 0 at peak, s = 0 when it lands if on flat ground).

Horizontal: Since there is no horizontal acceleration (ignoring air resistance), the horizontal motion is uniform: s = u×cos(θ) × t. The time t is the same as the time found from the vertical equations.

The kinematics solver handles each component separately. Run it first for the vertical component to find t, then use that t in the horizontal component to find the range.

Common Mistakes to Avoid

Using the wrong equation

Each SUVAT equation omits one variable. If you need to find s but you try to use v² = u² + 2as, you are fine — but if you misidentify which variable you are solving for, you may apply an equation that cannot be solved for the unknown. The solver handles equation selection for you.

Confusing displacement with distance

Displacement is a vector — it has direction. If an object goes 50 m forward then 30 m back, its displacement is +20 m but its distance travelled is 80 m. SUVAT uses displacement (s), not distance. For problems where the direction reverses, split into two separate SUVAT problems: one for each phase of motion.

Using g = 10 m/s² vs 9.81 m/s²

Many exam questions specify which value of g to use. If the question says g = 10 m/s², use 10. If it says "take g as 9.81 m/s²", use 9.81. The kinematics solver defaults to 9.81 m/s² but accepts any value you enter for acceleration.

Forgetting units

Always check that your inputs are in SI units (m, m/s, m/s², s) before using the solver. A speed in km/h must be converted to m/s by dividing by 3.6. A distance in km must be converted to m by multiplying by 1000.

Common Questions

Can I use the solver if I only know two variables?

No. The SUVAT equations have five variables; knowing two is not sufficient to determine the other three — the system is underdetermined. You need three known values to solve for the two unknowns. If a problem seems to give you only two values, look again — often the problem implies a third value (e.g., "starts from rest" means u = 0; "comes to a stop" means v = 0; "dropped" means u = 0 and a = 9.81 m/s²).

What if the solver gives two answers for time?

The equation s = ut + ½at² is quadratic in t, so it can have two solutions. Both may be physically valid: one is the time when the projectile passes through the height on the way up, the other is when it passes through on the way down. Choose the answer that fits the physical context — usually the positive value or the later time.

Does the solver handle units other than SI?

The solver works with whatever units you enter, but the equations assume internally consistent units. If you mix km/h and m/s within one problem, the answer will be incorrect. Convert all values to the same unit system before entering them.

How does kinematics relate to dynamics?

Kinematics describes motion without referring to the forces that cause it. Dynamics (Newton's second law, F = ma) brings in forces. Often you solve a dynamics problem first to find the acceleration, then use that acceleration in the kinematics solver to find displacement, velocity, or time. The two approaches complement each other.