CLASSICAL MECHANICS

Big Four Kinematics Calculator

Use the Big Four kinematic equations to solve constant-acceleration physics problems involving velocity, acceleration, displacement, and time. Enter the quantities your homework problem gives you and the solver will choose an equation, substitute the values, and walk you through the solution. This kinematics calculator works as a velocity calculator, acceleration calculator, displacement calculator, and free fall calculator — all in one place. Whether you need a quick kinematic equations calculator for a single problem or full step-by-step physics homework help, the tools below will guide you through how to solve kinematics problems.

PHYSICS HOMEWORK TOOL

Free Kinematics Homework Solver

Enter the quantities your problem gives you and leave the quantity you need to find blank. The solver will choose the right Big Four kinematic equation, substitute your values, and walk you through every step.

Free-Fall Mode

Initial Velocityv₀

m/s

Final Velocityv

m/s

Accelerationa

m/s²

Timet

DisplacementΔx

EQUATION SELECTOR

Which Kinematic Equation Should I Use?

Select what your problem gives you and what you need to find.

I know:

I need to find:

REFERENCE

The Four Kinematic Equations

1. v = v₀ + at

Velocity as a function of acceleration and time

2. Δx = v₀t + ½at²

Displacement as a function of initial velocity, acceleration, and time

3. v² = v₀² + 2aΔx

Velocity squared as a function of acceleration and displacement (no time needed)

4. Δx = ((v₀ + v) / 2)t

Displacement using average velocity (no acceleration needed)

Each equation relates exactly four of the five constant-acceleration variables. To solve a problem, you need three known quantities plus the one you are solving for — and all four must appear in the same equation. Use the Which Kinematic Equation Should I Use? tool above if you are not sure which one to pick.

INSTANT CALCULATOR

Three-Input Quick Solve

The four kinematic equations describe motion under constant acceleration. Enter an initial velocity, acceleration, and elapsed time to solve for final velocity and displacement.

Initial velocity (u) · m/s

Acceleration (a) · m/s²

Time (t) · s

FINAL VELOCITY (v)49 m/s

DISPLACEMENT (s)122.5 m

THE FOUR EQUATIONS

v = u + at

49 = 0 + 9.8 × 5

s = ut + ½at²

122.5 = 0×5 + ½×9.8×5²

v² = u² + 2as

2,401 = 0 + 2×9.8×122.5

s = ½(u + v)t

122.5 = ½(0 + 49)×5

Starting at 0 m/s and accelerating at 9.8 m/s² for 5 s, the body reaches 49 m/s after traveling 122.5 m.

CALCULUS FOUNDATIONS

Where the Big Four Come From

The Big Four kinematic equations come directly from the calculus definitions of velocity and acceleration, assuming constant acceleration. The starting point is:

v = dx/dt

a = dv/dt

where x is position, v is velocity, a is acceleration, and t is time.

Deriving v = v₀ + at

Start with acceleration:

a = dv/dt

If acceleration is constant, rearrange:

dv = a dt

Integrate both sides from the initial velocity v₀ at t = 0 to velocity v at time t:

∫_v₀^v dv = ∫₀^t a dt

Because a is constant:

v − v₀ = at

Therefore:

v = v₀ + at

This equation simply says that velocity changes linearly with time when acceleration is constant.

Deriving Δx = v₀t + ½at²

Velocity is:

v = dx/dt

From the first equation:

v = v₀ + at

Substitute that into the velocity definition:

dx/dt = v₀ + at

Rearrange:

dx = (v₀ + at) dt

Now integrate from the initial position x₀ to the final position x:

∫_x₀^x dx = ∫₀^t (v₀ + at) dt

Integrating gives:

x − x₀ = v₀t + ½at²

Since Δx = x − x₀, we get:

Δx = v₀t + ½at²

The first term represents displacement caused by the object's initial motion. The second represents the extra displacement produced by acceleration.

Deriving v² = v₀² + 2aΔx

This one is particularly interesting because it eliminates time.

Start with:

a = dv/dt

Using the chain rule:

dv/dt = (dv/dx)(dx/dt)

But dx/dt = v, therefore:

a = v (dv/dx)

Rearrange:

v dv = a dx

Now integrate:

∫_v₀^v v dv = ∫_x₀^x a dx

Since acceleration is constant:

½(v² − v₀²) = a(x − x₀)

Replace (x − x₀) with Δx:

½(v² − v₀²) = aΔx

Multiply by 2:

v² − v₀² = 2aΔx

Therefore:

v² = v₀² + 2aΔx

This is why this equation is so useful when time is not given.

Deriving Δx = ((v₀ + v) / 2) t

For constant acceleration, velocity changes linearly:

v = v₀ + at

The average velocity is therefore simply the average of the beginning and ending velocities:

v_avg = (v₀ + v) / 2

Displacement equals average velocity times time:

Δx = v_avg t

Substitute:

Δx = ((v₀ + v) / 2) t

We can also derive this directly with integration.

Displacement is:

Δx = ∫₀^t v dt

Using v = v₀ + at gives:

Δx = ∫₀^t (v₀ + at) dt

so:

Δx = v₀t + ½at²

From v = v₀ + at we can solve for:

at = v − v₀

Thus:

½at² = ½(v − v₀)t

Substitute:

Δx = v₀t + ½(v − v₀)t

Combine terms:

Δx = ½v₀t + ½vt

Therefore:

Δx = ((v₀ + v) / 2) t

The Deeper Idea

There are really only two fundamental calculus relationships behind all four equations:

v = dx/dt

a = dv/dt

The Big Four are just different consequences of integrating or rearranging those relationships under the special condition:

a = constant

That last condition matters. If acceleration changes with time, the standard Big Four generally cannot be used directly. Instead, you have to work with functions such as:

a(t)

and integrate:

v(t) = v₀ + ∫ a(t) dt

followed by:

x(t) = x₀ + ∫ v(t) dt

So calculus is not an alternative to the Big Four. Calculus is where the Big Four come from.

CRITICAL FOR STUDENTS

When NOT to Use the Big Four

The Big Four kinematic equations are powerful, but they work only under specific conditions. Their biggest limitation is that they assume constant acceleration. If acceleration changes during the motion, the standard Big Four usually cannot describe the entire motion with one equation.

Works Well

A car accelerating steadily at a = 3.0 m/s²

Does NOT Work

A car whose acceleration changes every second

Other Important Limitations

Constant acceleration only

If a varies with time, position, or velocity, calculus or numerical methods are usually required.

Usually applied one dimension at a time

For projectile motion, you must treat the horizontal and vertical directions separately.

They do not automatically account for air resistance

Standard free-fall problems assume drag is negligible.

They do not describe forces

The Big Four tell you how motion changes, but not why. To determine the cause of acceleration, you need Newton's laws.

They require a consistent reference frame and sign convention

Choosing upward as positive versus downward as positive changes the signs of velocity, displacement, and acceleration.

They assume ordinary classical mechanics

They are not appropriate when relativistic effects become important, such as motion near the speed of light.

They do not inherently handle curved paths

Circular or rotational motion usually requires additional equations involving angular velocity, centripetal acceleration, or vectors.

They can produce mathematically valid but physically irrelevant answers

For example, solving a quadratic for time may give a negative time or two positive times. You must interpret the result physically.

They depend on accurate initial conditions

If v₀, a, or displacement is incorrectly identified, the equation can produce a perfectly calculated but incorrect answer.

The Big Four describe a particular model of motion, not all motion.

They are excellent when acceleration is constant. Once acceleration varies, the more general relationships are:

a = dv/dt

v = dx/dt

Then you may need integration:

v(t) = v₀ + ∫ a(t) dt

x(t) = x₀ + ∫ v(t) dt

One of the most common mistakes students make is applying a kinematics equation to a situation where acceleration is not constant. Always check this first.