Introductory mechanics is a handful of equations that keep reappearing in different costumes. What makes them hard is rarely the algebra — it is knowing which one applies, and noticing when a quantity is squared, because that changes the answer’s behaviour completely.
Force: F = ma
Newton’s second law says force equals mass times acceleration. A newton is the force that accelerates one kilogram at one metre per second squared, which is roughly the weight of a small apple in your hand.
The word doing the work here is acceleration, meaning any change in velocity — speeding up, slowing down, or turning. A car going round a bend at constant speed is accelerating, because its direction is changing, and that is why it needs grip to do it.
The other half is that net force is what matters. A book resting on a table has gravity pulling it down and the table pushing it up; they cancel, the net force is zero, and it stays put. The force calculator solves for whichever of the three you leave out.
Kinetic energy, and why the square matters
KE = ½mv². Energy of motion, in joules. The square is the whole story: doubling the speed quadruples the energy.
This is the single most practically important fact in this guide. A car at 60 km/h carries four times the energy it does at 30, not twice, and all of it has to go somewhere in a crash. It is also why stopping distances grow so sharply with speed — the brakes have to dissipate energy that grew with the square.
Compare that with momentum, p = mv, which is linear. Two objects can have the same momentum and very different energies, which is why a slow lorry and a fast car are not interchangeable problems. Momentum is conserved in a collision; kinetic energy generally is not, because some becomes heat and deformation. The kinetic energy and momentum calculators sit either side of that distinction.
Work and power are not the same thing
W = Fd — work is force times the distance moved along that force. Carrying a heavy box across a level room does no work on the box in the physics sense, because the force is upward and the motion is horizontal. It is exhausting, but that energy goes into holding your muscles tense, not into the box.
Power is work divided by time. Lifting the same box up the same stairs is the same work whether you take five seconds or five minutes; the power differs by a factor of sixty. This is exactly what separates a watt from a joule, and the work done calculator reports both.
Because power and energy get quoted in so many different units — watts, horsepower, joules, calories, kilowatt-hours — the power and energy converters are often the faster route than doing it by hand.
Torque is force with leverage
τ = Fr. Torque is a twisting effect, and it depends on where you push as much as how hard. Doubling the spanner length doubles the torque for the same effort, which is the entire reason long spanners exist.
The subtlety is that r is the perpendicular distance from the pivot to the line of the force. Pushing along the spanner rather than across it produces no torque at all, however hard you push. The torque calculator converts into pound-feet as well, since fasteners are still specified that way in much of the world.
Springs: F = kx
Hooke’s law says the force a spring exerts is proportional to how far it is stretched or compressed. The constant k is the stiffness, in newtons per metre.
Stored energy, though, is ½kx² — squared again. Pulling a spring twice as far stores four times the energy, which is why the last centimetre of a drawn bow does so much more than the first.
Hooke’s law holds only up to the elastic limit. Past that the material deforms permanently and the equation stops describing anything real, which is a genuine limit rather than a technicality. The Hooke’s law calculator gives the stored energy alongside the force.
Projectiles: two problems, not one
The trick that makes projectile motion tractable is that the horizontal and vertical motions are independent. Horizontally, nothing accelerates it, so it travels at constant speed. Vertically, gravity acts at about 9.81 m/s² regardless of what it is doing sideways.
The famous consequence: a bullet fired horizontally and one dropped from the same height hit the ground at the same moment. Their vertical stories are identical, and the horizontal motion has no say in it.
On level ground, maximum range comes at 45 degrees, and angles either side of it pair up — 30 and 60 degrees give the same distance by different routes, one flatter and faster, one higher and slower. The projectile motion calculator gives range, flight time and peak height together. All of this assumes no air resistance, which is fine for a thrown ball and badly wrong for anything light or very fast.
Gravity between any two masses
F = Gm₁m₂/r². Every mass attracts every other, with G at 6.674 × 10−11 — a tiny number, which is why you feel no pull toward the person next to you.
The r² is an inverse square: double the distance and the force drops to a quarter, not a half. And r is measured centre to centre, so for anything sitting on Earth’s surface it is the planet’s radius — which is why climbing a mountain barely changes your weight. The gravitational force calculator handles the very large and very small numbers involved.
The habits that prevent most mistakes
Convert to SI before substituting — kilograms, metres, seconds — and most unit errors disappear on their own. Check whether the quantity you want is squared, because that decides whether doubling an input doubles or quadruples the answer. And check the result is the right size: if a thrown ball comes out with the kinetic energy of a truck, the mistake is upstream, and finding it is faster than redoing the algebra.