Free Mechanics Calculator with Steps

One physics mechanics calculator, three topics

Most introductory mechanics problems come from three small families of formulas: motion with constant acceleration (kinematics), Newton's second law (force), and work and energy. This physics mechanics calculator keeps all three in one place, and each one can be solved for any of its variables — not just the one it is usually written for.

What makes a mechanics answer trustworthy is not the final number but the chain that produced it: which equation you chose, what you substituted, and whether the units survived the algebra. That is why every result here is rendered as formula → substitution → result, with SI units attached at the end. If your homework answer differs, comparing line by line shows exactly where the two calculations split.

• v = u + at — final velocity from initial velocity, acceleration, and time
• s = ut + ½at² — displacement covered under constant acceleration
• v² = u² + 2as — links the two velocities to displacement when time is unknown
• F = ma — Newton's second law: net force, mass, and acceleration
• W = Fd — work done by a force acting along the direction of motion
• KE = ½mv² — kinetic energy of a moving mass
• PE = mgh — gravitational potential energy relative to a chosen reference height

How to choose a kinematics equation

The three constant-acceleration equations use five quantities between them — initial velocity u, final velocity v, acceleration a, time t, and displacement s — but each equation contains only four. That is the whole selection trick: v = u + at has no s in it, s = ut + ½at² has no v, and v² = u² + 2as has no t. List what the problem gives you and what it asks for, then pick the equation that leaves out the one quantity you neither know nor need.

For example, "a car brakes from 20 m/s to rest over 50 m — find the deceleration" gives u, v, and s and asks for a. Time never appears, so v² = u² + 2as is the equation, rearranged to a = (v² − u²) / 2s = (0 − 400) / 100 = −4 m/s². The negative sign is not an error; it says the acceleration points against the motion, which is exactly what braking means.

Two caveats keep this kinematics calculator honest. First, all three equations assume acceleration is constant — they do not apply to a force that changes with time or position, such as air resistance near terminal velocity. Second, u, v, a, and s are signed quantities along one chosen axis. Pick a positive direction before substituting and keep it: for an object thrown upward with "up" positive, gravity enters as a = −9.81 m/s².

Force, work, and energy: F = ma, W = Fd, KE = ½mv², PE = mgh

Newton's second law relates the net force on an object to the acceleration it produces: F = ma. The F here is the sum of all forces acting, not any single one — a 700 N engine force against 200 N of friction accelerates the car with the 500 N difference. The unit works out from the formula itself: one newton is the force that gives one kilogram an acceleration of one metre per second squared, so N = kg·m/s². Used as a force calculator, the same equation also answers the inverse questions: m = F / a recovers the mass from a measured force and acceleration, and a = F / m predicts the acceleration a known force will produce.

Work is energy transferred by a force: W = Fd when the force acts along the direction of motion, measured in joules (1 J = 1 N·m). If the force acts at an angle θ to the motion, only its component along the motion does work, and the general form is W = Fd·cos θ — this calculator uses the parallel case, so resolve an angled force into its along-track component first. Kinetic energy is the energy of motion, KE = ½mv², and because speed is squared, doubling v quadruples the kinetic energy — the reason stopping distances grow so much faster than driving speed.

Gravitational potential energy near the Earth's surface is PE = mgh, with g ≈ 9.81 m/s². The h is height above a reference level you choose — the floor, a table top, the bottom of a hill — so a potential energy value only means something together with its reference. These formulas connect through conservation of energy: the work done lifting a mass becomes its potential energy, and the potential energy a falling mass loses reappears as kinetic energy, which is a fast way to cross-check answers between the tabs.

Common mistakes and quick checks

• Mixed units. The formulas assume SI throughout: convert km/h to m/s by dividing by 3.6, grams to kilograms, and centimetres to metres before substituting — 60 km/h entered as "60" with metres elsewhere is the single most common wrong answer.
• Using weight where the formula wants mass. A "5 kg" object is a mass; its weight is mg ≈ 49 N. If a problem hands you a weight in newtons, divide by g to recover the mass before using F = ma or KE = ½mv².
• Dropping the sign of acceleration. Deceleration is negative acceleration along your positive axis. Substituting a = 4 instead of a = −4 m/s² into s = ut + ½at² gives a longer distance, not a braking one.
• Forgetting the ½. It appears in both s = ut + ½at² and KE = ½mv², and omitting it exactly doubles the answer — an error worth recognising on sight.
• Applying constant-acceleration equations to non-constant forces. If the problem says the force varies, the SUVAT equations do not apply; energy methods often still do.
• Rounding too early. Carry full precision through intermediate steps and round only the final answer; squaring an early-rounded velocity amplifies the error.
• Check the result against a limiting case: with a = 0, s = ut + ½at² must collapse to s = ut, and v = u + at must return u. A quick unit check helps too — if your "energy" came out in kg·m/s, a square or a multiplication went missing.