What is classical mechanics?
Classical mechanics is the physics of motion and force for objects much more massive than atoms, much slower than light, and experiencing only everyday (non-quantum, non-relativistic) interactions. Three quantities carry everything in this domain: position (where), velocity (how fast, in what direction), and force (what pushes). The calculus builds acceleration, momentum, energy, work, and power from those three. The six calculators cover the working toolkit: projectile motion (parabolic flight under constant gravity), free fall (vertical drop from rest), inclined plane (gravity split into parallel and normal components plus friction), Newton's 2nd law (force, mass, acceleration, weight), momentum & impulse (the rate of change of motion), and work, energy, and power (the conserved bookkeeping).
Four conserved quantities do the heavy lifting: momentum (p = mv) when no external force acts; mechanical energy (KE + PE) when only conservative forces act; angular momentum when no external torque acts; mass in ordinary (non-nuclear) interactions. Together they solve almost every textbook problem.
Projectile motion — the parabolic flight
Launch anything with initial speed v0 at angle θ above horizontal and the trajectory becomes a parabola when air resistance is negligible. The horizontal and vertical motions are independent: x(t) = v0 cosθ · t (constant horizontal speed) and y(t) = v0 sinθ · t − ½g t² (uniformly accelerated vertical drop). Three summary results follow from those kinematics: range R = v0² sin(2θ) / g, max height H = v0² sin²(θ) / (2g), time of flight T = 2v0 sinθ / g (for a launch from ground level). The default sample (v0 = 20 m/s, θ = 45°, Earth g = 9.81 m/s²) gives vx = vy = 14.142 m/s, R = 40.775 m, H = 10.194 m, and T = 2.883 s. The 45° angle gives the maximum range on level ground for a given speed , and on the Moon the same launch reaches R = 246.91 m because gravity is six times weaker.
Free fall — three equivalent kinematic forms
An object released from rest falls under constant gravity with three interchangeable relations: v = gt (instantaneous speed at time t), h = ½g t² (distance fallen from rest after time t), and v = √(2gh) (speed after falling distance h). Each is a special case of the kinematic equations and convertible to the others by substitution. The default sample (h = 100 m, Earth) gives t = 4.515 s, v = 44.294 m/s, and v = 159.46 km/h — roughly terminal velocity for a skydiver in a stable spread-eagle position. Switch to from time mode and a 3-second drop yields d = 44.145 m and v = 29.43 m/s (about 105.95 km/h). The same drop on the Moon takes t = 11.11 s and reaches only 17.97 m/s — the same fall, six times slower.
Inclined plane — gravity split into components
A block of mass m on a slope of angle θ feels gravity W = mg straight down, but only the component along the surface accelerates the block; the component into the surface is cancelled by the normal force. The split is geometry: Fparallel = mg sinθ, normal N = mg cosθ. Friction adds a resisting force f = μN = μmg cosθ opposing motion; the kinetic friction coefficient μ ranges from about 0.04 (steel on ice) to 0.8 (rubber on rubber). The net down-slope force is Fnet = mg sinθ − μmg cosθ, giving acceleration a = g(sinθ − μ cosθ). The default sample (m = 10 kg, θ = 30°, μ = 0.2, g = 9.81 m/s²) yields W = 98.1 N, N = 84.957 N, Fparallel = 49.05 N, f = 16.991 N, Fnet = 32.059 N, and a = 3.206 m/s². The block slides down because the parallel component exceeds friction. Crank μ up to 0.6 and the net force becomes negative — the block stays put (or slides back up if you pushed it); friction has won. The angle at which a frictional block first slides is arctan(μ), the friction angle; below it, static friction holds the block.
Newton's 2nd Law — F = m · a, three solve-for modes
Newton's 2nd law in its scalar form is F = m · a: the net force on an object equals its mass times its acceleration. The default (mass 10 kg, acceleration 5 m/s²) yields F = 50 N. Flip the dropdown and the same panel solves for mass (m = F / a), acceleration (a = F / m), or weight (W = m · g, using Earth gravity 9.81 m/s²). A 70 kg person has W = 686.7 N on Earth and W = 113.4 N on the Moon — the same person, the same mass, but a sixfold weight difference because gravity is six times weaker. Weight is a force (newtons); mass is an amount of matter (kilograms).
Momentum and impulse — the twin conservation laws
An object in motion carries momentum p = mv, a vector pointing along the velocity. To change an object's momentum requires impulse J = FΔt = Δp, equal to the force applied over the time of application. The default sample (m = 2 kg, v = 15 m/s, F = 50 N, t = 0.5 s) yields p = 30 kg·m/s, KE = 225 J, J = 25 N·s, Δv = 12.5 m/s, and a final velocity of vfinal = 27.5 m/s if the impulse was applied in the direction of motion. The same impulse delivered by a smaller force over a longer time (a "soft" catch) or a larger force over a shorter time (a "hard" collision) gives the same momentum change. In a closed system with no external forces, total momentum is conserved: a 2 kg object at 15 m/s hitting a stationary 1 kg object ends with v1' = 5 m/s and v2' = 20 m/s (perfectly elastic) or both share a common velocity v = 10 m/s (perfectly inelastic). That bookkeeping is the basis of every collision analysis from billiards to proton beams.
Work, energy, and power — the conserved bookkeeping
When a force moves an object through a distance, it does work W = F · d (the dot product makes it exact for off-axis forces: only the component of F along d counts). The energy an object has by virtue of its motion is kinetic energy KE = ½mv²; the energy it has by virtue of its position in a gravitational field is potential energy PE = mgh. The total mechanical energy E = ½mv² + mgh is conserved when only gravity acts (no friction, no applied forces) — which is why a pendulum swings to the same height on every cycle, and why a ball thrown upward returns to its launch speed on the way down. Power P = W / t measures the rate at which work is done. The default sample (m = 5 kg, v = 10 m/s, h = 20 m, g = 9.81, F = 50 N, d = 4 m, t = 2 s) yields KE = 250 J, PE = 981 J, total mechanical energy = 1231 J, W = 200 J, P = 100 W, and P = 0.1341 hp.
Common misconceptions
- Heavier objects fall faster than lighter ones. In a vacuum, the opposite is true: every object accelerates at the same g regardless of mass. Air resistance breaks the tie because it scales with cross-section, not weight — a feather and a hammer dropped together on Earth hit the ground at different times, but on the Moon (no air) they hit together, demonstrated live during Apollo 15.
- Force and velocity are the same thing. Force causes acceleration (F = ma); velocity is what you get from sustained force. A stalled car has force at the wheels but zero velocity; a hockey puck slides across frictionless ice at constant velocity with zero net force.
- Momentum and kinetic energy are the same. Both vanish at rest, but they scale differently. Doubling the velocity doubles the momentum (p = mv) but quadruples the kinetic energy (KE = ½mv²). A 2-ton truck at 60 mph has the same momentum as a 1-ton car at 120 mph but twice the kinetic energy.
- "g" is a force. g is the acceleration due to gravity (m/s²). The force on an object from gravity is W = mg. A 1 kg mass weighs 9.81 N on Earth; on the Moon it weighs 1.62 N because g is six times smaller.
Related tools: Vector Calculator for decomposing forces and velocities into components, Electricity & Circuits for the electromagnetic force context, Waves & Oscillations for the oscillating cousins of projectile motion, Rotation & Torque for the angular-momentum counterpart of straight-line motion, and Thermodynamics & Heat for the energy side of mechanics.
