Newton's First Law

The law

An object at rest stays at rest. An object in motion continues at constant velocity (constant speed in a straight line). Unless a net external force acts on it.

Mathematically:

ΣF = 0  ⟹  a = 0  ⟹  v = constant

The "net external force" matters — internal forces (e.g., parts of an object pulling on each other) can't change the object's overall motion. Only external forces from the environment can.

Why this overturned Aristotle

For 2,000 years, Aristotle's view dominated — objects in motion naturally come to rest. Motion required a sustained "mover." Watching a horse pulling a cart, this seems obvious — stop pulling, the cart stops.

Galileo (1564-1642) ran inclined-plane experiments showing balls roll farther on smoother surfaces. Extrapolating to a perfectly frictionless surface, a ball would roll forever. Galileo concluded motion is preserved, not "naturally" lost. Friction is the cause of stopping, not the absence of a force.

Newton's Principia (1687) formalized this insight as the first law. The "natural state" of objects became constant velocity, not "at rest" — a profound conceptual shift.

Inertial reference frames

The first law only holds in inertial reference frames — frames where there are no fictitious forces. Examples:

Frame	Inertial?	Notes
A stationary lab on Earth	Approximately	Earth's rotation gives small Coriolis force; usually ignored at small scales
A car at constant 60 mph on straight road	Yes	Constant velocity, no acceleration
A car braking	NO	Decelerating; loose objects "fly forward" — apparent fictitious force
A rotating carousel	NO	Centripetal acceleration; "centrifugal force" appears in this frame
A free-falling elevator	Yes (locally)	Equivalence principle — feels like floating
Earth orbiting Sun	NO (approximately yes for small experiments)	Centripetal acceleration toward Sun
Free-floating spacecraft (no thrust)	Yes	Best practical inertial frame

Inertia and mass

Inertia is an object's resistance to changes in motion. The more mass, the more inertia. Pushing a shopping cart to start it moving is easy; pushing a car requires much more force for the same change in velocity.

Mathematically (from F = ma), the same force produces less acceleration for a more massive object. Mass is a quantitative measure of inertia.

Note — mass is NOT the same as weight. Mass is intrinsic (kg). Weight is the gravitational force on mass (N = mass × g). On the Moon, your weight is 1/6 of Earth's, but your mass is unchanged. Inertia depends on mass, not weight.

Real-world consequences

Situation	Inertia at work
Seatbelts	In a sudden stop, your body continues at original velocity; belt applies decelerating force
Airbags	Spread the deceleration over time; reduce force on body (impulse-momentum)
Tablecloth trick	Quick yank — too brief to overcome dishes' inertia; dishes stay roughly at rest
Magic-trick drop	A coin balanced on a card; flick the card → card moves, coin drops into glass (inertia keeps it nearly in place)
Spacecraft cruise	Once on trajectory in deep space, no thrust needed — motion continues
Heavy doors	Hard to start swinging, hard to stop — large mass means large inertia

JavaScript — simulating inertia

// Object with no force applied — should keep moving at constant velocity
class Particle {
  constructor(x, y, vx, vy) {
    this.x = x; this.y = y;
    this.vx = vx; this.vy = vy;
  }
  
  step(dt, fx = 0, fy = 0, mass = 1) {
    // a = F/m
    const ax = fx / mass;
    const ay = fy / mass;
    // v += a·dt
    this.vx += ax * dt;
    this.vy += ay * dt;
    // x += v·dt
    this.x += this.vx * dt;
    this.y += this.vy * dt;
  }
}

// Test: no force → constant velocity
const p = new Particle(0, 0, 5, 0);  // moving 5 m/s right
for (let t = 0; t < 10; t++) {
  p.step(1);  // 1 second steps
}
console.log(p.x, p.y);  // 50, 0 — moved 50m, no change in velocity ✓
console.log(p.vx, p.vy); // 5, 0 — velocity unchanged

// With friction (kinetic): F_friction = -μ·m·g·v̂ (opposing motion)
function withFriction(mass, vx, vy, mu = 0.1, g = 9.81, dt = 0.1) {
  const speed = Math.sqrt(vx * vx + vy * vy);
  if (speed < 1e-9) return [0, 0];
  // Friction force opposes velocity
  const fx = -mu * mass * g * (vx / speed);
  const fy = -mu * mass * g * (vy / speed);
  return [fx, fy];
}

// Without friction, p moves forever; with it, p decelerates per Newton's 2nd

When the law applies (and doesn't)

• Holds in inertial frames. Earth labs, cars at constant velocity, free-falling spacecraft.
• Apparently violated in non-inertial frames. In a braking car, your coffee "lurches forward" — but no real force pushed it. The car is decelerating; your coffee is following Newton's first law (continuing at original velocity) while the car decelerates around it.
• Galilean — not relativistic. Newton's laws are accurate for v ≪ c. At relativistic speeds, momentum p = γmv (not mv); time dilates; lengths contract. Special relativity replaces Newton's framework but agrees in the low-speed limit.
• Classical — not quantum. Quantum particles have momentum and energy uncertainty (Heisenberg); the deterministic "constant velocity" description fails at atomic scales.

Common mistakes

• Confusing inertia with mass. Inertia is the property; mass is the quantity. They're tightly linked but not synonymous. "More mass = more inertia" is the correct phrasing.
• Thinking "rest" is special. An object at rest and one moving at constant velocity are equivalent — both have ΣF = 0. The first law treats them identically (in their respective inertial frames).
• Forgetting "net" force. A book on a table has gravity and normal force, both nonzero — but they cancel. ΣF = 0 still holds; book stays at rest.
• Applying it in non-inertial frames. In a rotating frame, objects appear to accelerate without external forces (centrifugal, Coriolis). The first law fails — you need fictitious forces or move to an inertial frame.
• Treating "uniform motion" as "no motion." A spacecraft cruising at 30 km/s through deep space has no net force on it (Newton's first law) — but it's NOT at rest. Uniform velocity ≠ zero velocity.
• Misattributing "stopping forces." A rolling ball stops because of friction and air drag, not because motion "naturally ends." This was Aristotle's mistake.