Newton's Third Law

The law

For every force, there is an equal and opposite force.

If object A exerts force F on object B, then object B simultaneously exerts force −F on object A. The forces are equal in magnitude and opposite in direction.

F_AB = −F_BA

The crucial point — these forces act on DIFFERENT objects. They don't cancel out (they're not on the same object).

Action-reaction pairs everywhere

Action	Reaction	Result
Foot pushes ground backward	Ground pushes foot forward	You walk forward
Rocket pushes exhaust gas backward	Gas pushes rocket forward	Rocket accelerates
Bullet fired forward	Gun recoils backward	Recoil felt by shooter
Earth pulls apple down	Apple pulls Earth up	Both accelerate (Earth's tiny)
Tire pushes road backward	Road pushes tire forward	Car accelerates forward
Swimmer pushes water back	Water pushes swimmer forward	Swimmer moves forward
Helicopter blade pushes air down	Air pushes blade up	Helicopter rises
Hammer hits nail (force on nail)	Nail pushes hammer back	Hammer decelerates after impact

Why action-reaction doesn't cancel motion

The most common confusion — "if forces are equal and opposite, why does anything move?"

Answer — they act on different objects. Each object has its own ΣF = ma equation:

• You push a heavy stalled car with force F. Car gets force F (forward), accelerates a tiny bit.
• By Newton's third, car pushes you back with force F (backward). YOU get force F backward.
• But you also have friction from your feet on the road, pushing you forward.
• Net force on you: friction − reaction = small or zero. You don't accelerate.
• Net force on car: only your push (no friction balancing). Car accelerates.

So both forces are real, equal in magnitude, but applied to different objects with different other forces.

Rockets and propulsion

The third law is the foundation of all propulsion:

System	What's pushed back	What's pushed forward
Rocket	Exhaust gas (high velocity)	Rocket body
Jet engine	Air (accelerated through engine)	Aircraft
Propeller (boat/airplane)	Water/air (pushed back)	Vehicle
Walking	Ground (foot pushes it back)	Person (ground pushes back)
Swimming	Water (pushed by hands/feet)	Swimmer
Squid/octopus	Water (jetted out)	Squid (forward)

Note — rockets work in vacuum (no air to push against) precisely because the law says the rocket and exhaust push on EACH OTHER. No external medium is needed.

Momentum conservation as a consequence

If forces always come in equal-opposite pairs, then internal forces in an isolated system can't change the system's total momentum. Here's why:

For any two objects A and B in a system, F_AB = −F_BA. Over time dt, A's momentum changes by F_AB·dt. B's changes by F_BA·dt = −F_AB·dt. Total change: F_AB·dt + (−F_AB·dt) = 0.

This holds for every pair, so the total momentum of an isolated system never changes. Momentum conservation is a direct consequence of Newton's third law.

JavaScript — collision via 3rd law

// Two-body interaction: equal-opposite forces, momentum conserved
class Body {
  constructor(mass, x, vx) {
    this.mass = mass;
    this.x = x;
    this.vx = vx;
  }
}

// Apply equal-opposite forces over dt
function step(a, b, F_AB, dt) {
  // Force on A from B is F_AB; force on B from A is -F_AB
  a.vx += (F_AB / a.mass) * dt;
  b.vx += (-F_AB / b.mass) * dt;
  a.x += a.vx * dt;
  b.x += b.vx * dt;
}

// Verify momentum conservation
function totalMomentum(bodies) {
  return bodies.reduce((sum, b) => sum + b.mass * b.vx, 0);
}

const ball1 = new Body(2, 0, 5);
const ball2 = new Body(3, 5, -3);
console.log('Initial momentum:', totalMomentum([ball1, ball2])); // 2*5 + 3*(-3) = 1

// Apply 100 N from 2 onto 1 for 0.1 second (so 100 N from 1 onto 2 in reverse)
step(ball1, ball2, 100, 0.1);
console.log('After force pair:', totalMomentum([ball1, ball2])); // Still 1 ✓

// Rocket simulation: thrust = exhaust mass flow × velocity
function rocketStep(rocket, dt, exhaustMassRate, exhaustVelocity) {
  // Force on rocket = +exhaustMassRate × exhaustVelocity (forward)
  // Equal opposite on exhaust (we don't track exhaust here)
  const thrust = exhaustMassRate * exhaustVelocity;
  rocket.vx += (thrust / rocket.mass) * dt;
  rocket.x += rocket.vx * dt;
  rocket.mass -= exhaustMassRate * dt; // mass decreases
}

const rocket = new Body(1000, 0, 0);
for (let t = 0; t < 60; t++) {
  rocketStep(rocket, 1, 10, 2500); // burn 10 kg/s, exhaust at 2500 m/s
}
console.log(`After 60s: v = ${rocket.vx.toFixed(0)} m/s, m = ${rocket.mass.toFixed(0)} kg`);

Identifying paired forces

Always ask — what's the action, and what's the reaction?

Force	Paired with
Earth's gravity on you (your weight)	Your gravity on Earth (yes — you pull Earth!)
Floor's normal force on you	Your weight pressing down on floor
Air resistance on a falling skydiver	Skydiver pushing air down
Spring pulling on you	You pulling on spring
Magnet attracting iron	Iron attracting magnet (yes — both ways)
Sun's gravity on Earth	Earth's gravity on Sun (Earth slightly tugs Sun)

Note — gravity on you from Earth is paired with YOUR gravity on Earth, NOT with the normal force from the floor. The normal force is a different action-reaction pair (you push down on floor; floor pushes up on you).

Where Newton's third law shows up

• Propulsion. Rockets, jets, propellers, walking, swimming, squid jets — all third-law applications.
• Recoil. Firearms, cannons, sneeze backward force, recoil from blocks of firework launching.
• Collision physics. All collisions involve action-reaction pairs; this is why momentum is conserved in collisions.
• Engineering — friction-based locomotion. Cars, trains, walking robots — all rely on friction (action-reaction with ground).
• Astrodynamics. Tidal forces (Earth pulls Moon, Moon pulls Earth), gravitational interactions in N-body systems.
• Sports — every sport. Pushing off the ground (running), pushing water (swimming), pushing air (cycling), pushing the ball (throwing).
• Detecting non-inertial frames. Real forces have third-law partners; fictitious forces (centrifugal, Coriolis) don't. If you can't find a partner, you might be in a non-inertial frame.

Common mistakes

• Thinking action-reaction pairs cancel. They act on DIFFERENT objects, so they don't cancel — each object analyzes its own ΣF separately.
• Pairing the wrong forces. Gravity on you (weight) pairs with YOUR gravity on Earth, not with normal force from floor. The normal force pair is your push on the floor.
• Forgetting that lighter objects accelerate more. Action-reaction forces are equal in MAGNITUDE, but accelerations are F/m. A bullet's acceleration is huge; the gun's is much smaller. Equal forces, very different accelerations.
• Ignoring third-law forces in dynamics. When the apple falls toward Earth, Earth also "falls" toward the apple — by an immeasurably tiny amount, but technically. For two-body problems (binary stars, Earth-Moon), this matters.
• Applying it to fictitious forces. Centrifugal and Coriolis don't have third-law partners. They appear in non-inertial frames as bookkeeping devices.
• Confusing 3rd law with equilibrium. 3rd law says forces come in pairs (always true). Equilibrium says ΣF = 0 on a single object (only true at rest or constant velocity). Different concepts.