Classical mechanics rests on a small set of formulas that describe how objects move and interact. Master these and a huge range of physics problems — from a falling ball to a braking car — become approachable. This guide walks through them in a logical order, showing how each one connects to the next.
Velocity and acceleration
Velocity is speed in a given direction, which is why a car going round a bend at constant speed is still accelerating — its direction is changing. When velocity changes, the object accelerates. The basic relation is v = u + at, where u is the starting velocity, a the acceleration and t the time. Acceleration itself is the change in velocity over time, a = (v − u) ÷ t, measured in metres per second squared. The velocity calculator and acceleration calculator handle both.
The equations of motion
For motion under constant acceleration, a family of equations (often called the suvat equations) link displacement, initial and final velocity, acceleration and time. Besides v = u + at, you can find distance with s = ut + ½at², and relate velocities and distance with v² = u² + 2as. Together these let you solve almost any constant-acceleration problem — how far a car travels while braking, or how high a ball thrown upward will rise — by picking the equation that contains the quantities you know and the one you want.
Gravity and free fall
Near the Earth's surface, gravity accelerates falling objects at about 9.8 m/s², regardless of their mass — famously, a feather and a hammer fall together in a vacuum. This single value plugged into the equations of motion predicts how long something takes to fall and how fast it lands. Air resistance complicates real falls, but for many problems the constant-acceleration model is an excellent approximation.
Force and momentum
Newton's second law states that force equals mass times acceleration: F = ma. A larger mass or a greater acceleration needs more force, measured in newtons. Closely related is momentum, the quantity of motion, given by p = mv. Momentum is conserved in collisions — the total before equals the total after — which is why it is central to analysing impacts, from snooker balls to car crashes. Use the force calculator and momentum calculator.
Work and energy
Work is done when a force moves an object: W = F × d, measured in joules. The energy of a moving object is its kinetic energy, ½mv², and the energy stored by a raised object is its gravitational potential energy, mgh. The great unifying principle is the conservation of energy: energy is never created or destroyed, only transformed. A falling object converts potential energy into kinetic energy; a car's brakes turn kinetic energy into heat. The work calculator, kinetic energy calculator and potential energy calculator cover these.
Power
Power measures how quickly work is done: P = W ÷ t, in watts. Two engines might do the same total work, but the more powerful one does it in less time — which is what we mean when we call a car powerful. The power calculator works it out from work and time.
A worked example
Drop a 2 kg ball from a height of 5 metres. Its potential energy at the top is mgh = 2 × 9.8 × 5 = 98 joules. By the moment it lands, all of that has become kinetic energy, so ½mv² = 98, giving a speed of about 9.9 m/s. Notice we found the landing speed using energy alone, without timing the fall — often there are two valid routes to an answer, and choosing the easier one is a skill in itself.
How it all connects
These formulas form a chain. A force accelerates a mass (F = ma); acting over a distance it does work (W = Fd); that work becomes kinetic or potential energy; and the rate of doing it is power. Energy and momentum are conserved, giving you shortcuts that bypass the details of the motion. The single habit that prevents most errors is keeping units consistent — kilograms, metres and seconds throughout. Get the units right, pick the equation that matches what you know, and the rest is arithmetic.
A simple problem-solving routine
When a mechanics problem looks daunting, a fixed routine tames it. First, list what you are given and what you are asked for, with units. Second, draw a quick sketch and mark the forces or motion. Third, choose the principle that links your knowns to your unknown — a force question points to F = ma, a speed-and-distance question to the equations of motion, and an 'how fast does it land' question often to energy conservation. Fourth, rearrange the formula before plugging in numbers, so the algebra stays clean. Finally, check that the answer's size and units make sense. This method works from school problems to real engineering, because the underlying laws are always the same handful you have met here.