Impulse

Definition

Impulse is the change in momentum of an object:

J = Δp = m·Δv

From Newton's second law (F = dp/dt), integrating over time gives:

J = ∫F dt = Δp

For constant force:

J = F·Δt

Units: N·s, equivalent to kg·m/s (the same as momentum). Impulse is a vector — direction matters.

The key insight — Δp = F·Δt

Two ways to give an object the same momentum change:

• Big force, short time. Hammer driving a nail. Bullet penetrating wood. Rear-ender at 30 mph into a brick wall.
• Small force, long time. Pushing a stalled car. Slow steady cooking with low heat. Decelerating a moving vehicle gradually with brakes.

Both have the same impulse. But the first hurts a lot; the second doesn't. Why? Peak FORCE matters for damage, not impulse.

Why airbags work — extending Δt

In a 30 mph head-on collision (Δv ≈ 13.4 m/s), your 70 kg body needs Δp = 70 × 13.4 = 938 kg·m/s of decelerating impulse.

Scenario	Δt	F_avg	Outcome
Head into windshield (no belt, no airbag)	~1 ms	938,000 N (~96,000 kg force)	Lethal trauma
Head/torso into seatbelt only	~100 ms	9,380 N (~960 kg force)	Bruising, possible chest injury
Head/torso into airbag + seatbelt	~300 ms	3,127 N (~320 kg force)	Mild discomfort, survivable
Same crash with high-end safety + crumple	~500+ ms	~1,876 N	Often walk away

By extending Δt by 300×, peak force drops by 300×. Your body can absorb ~6,000 N of localized force; ~1,000,000 N rips it apart. This factor of 300 is the difference between life and death.

Other impulse applications

Situation	Why impulse matters
Pitcher's follow-through	Extends contact time → more impulse → faster pitch
Tennis serve	Racquet drags through ball; longer contact = more energy transfer
Catching a ball	Pull hand back as ball arrives → extends Δt → reduces hand pain
Diver entering pool	Splayed entry: short Δt, painful. Streamlined entry: long Δt through water resistance, painless
Karate chop	Hard board, short Δt — high peak F → board breaks. Soft pillow doesn't break: long Δt, low F
Golf club head	Stiff face for short contact (more transmitted F); soft face deforms (longer Δt, more energy)
Bicycle helmet	Foam crumples on impact — extends Δt for the head, reduces peak force
Bungee jump	Long elastic cord stretches → long Δt of deceleration → comfortable landing

Rocket impulse and "specific impulse"

Rocket engines are characterized by specific impulse (Isp) — impulse per unit mass of propellant:

Isp = J / (m_propellant · g) = v_exhaust / g

Higher Isp means more momentum per kilogram of fuel — fewer trips to refuel.

Engine type	Isp	v_exhaust
Solid rocket boosters	~250 s	~2,500 m/s
Liquid hydrogen/oxygen	~450 s	~4,400 m/s
Ion thruster (Xenon)	~3,000 s	~30,000 m/s
VASIMR (concept)	~5,000-30,000 s	~50-300 km/s
Photonic (theoretical)	~30 million s	c (3 × 10⁸ m/s)

JavaScript — impulse and force-time

// Average force from impulse and time
function avgForce(deltaP, deltaT) {
  return deltaP / deltaT;
}

// Crash analysis
function crashForce(mass, vBefore, vAfter, contactTime) {
  const deltaP = mass * (vBefore - vAfter); // momentum change
  return avgForce(deltaP, contactTime);
}

// 70 kg passenger, 30 mph (13.4 m/s) → 0
console.log(`No airbag (1 ms): ${crashForce(70, 13.4, 0, 0.001).toFixed(0)} N`);
console.log(`With airbag (100 ms): ${crashForce(70, 13.4, 0, 0.1).toFixed(0)} N`);
console.log(`With airbag + crumple (300 ms): ${crashForce(70, 13.4, 0, 0.3).toFixed(0)} N`);

// Numerical integration of force over time → impulse
function impulse(forceArray, dt) {
  // Trapezoidal rule on F(t)
  let total = 0;
  for (let i = 1; i < forceArray.length; i++) {
    total += 0.5 * (forceArray[i-1] + forceArray[i]) * dt;
  }
  return total;
}

// Example: rectangular force profile (constant force F over time T)
function rectangularImpulse(F, T, dt = 0.001) {
  const N = Math.floor(T / dt);
  return impulse(Array(N).fill(F), dt);
}

console.log(rectangularImpulse(100, 0.5)); // 50 N·s

// Specific impulse from exhaust velocity
function specificImpulse(vExhaust, g = 9.81) {
  return vExhaust / g;
}

console.log(`Solid rocket Isp: ${specificImpulse(2500).toFixed(0)} s`);
console.log(`Hydrolox Isp: ${specificImpulse(4400).toFixed(0)} s`);

Where impulse shows up

• Auto safety. Airbags, crumple zones, seatbelts — all extend Δt to reduce peak force. Federal Motor Vehicle Safety Standards specify maximum allowable forces.
• Sports. Follow-through in throwing/hitting; cushion landing in catching; soft landing in jumps; bat/racquet design.
• Helmets and protective gear. Foam padding extends impact time. Football helmets have multi-layer foam for various impact severities.
• Rocket science. Specific impulse (Isp) is the figure of merit — high Isp means efficient propellant use.
• Particle physics. Detector design uses controlled impulse-energy relationships to identify particles.
• Construction safety. Catch nets, harnesses, bungee cords for fall protection — all designed to extend Δt.
• Hammer/hammer-anvil. Brittle materials snap from short, sharp impulses; ductile materials deform from longer, gentler impulses.

Common mistakes

• Confusing impulse with force. Impulse is total momentum change (a vector quantity). Force is rate of momentum change (also a vector). They have different units (N·s vs N).
• Ignoring Δt's role in safety. Big airbags work not by being soft but by extending Δt by orders of magnitude.
• Using F·Δt when force varies. If force changes over time, J = ∫F dt (the integral). Average force F_avg = Δp/Δt is correct, but using peak force times duration overstates impulse.
• Treating impulse and energy as interchangeable. They're different. A 1 kg ball at 10 m/s has p = 10 kg·m/s and KE = 50 J. Doubling speed: p = 20, KE = 200 (4×). Different scaling.
• Forgetting impulse is a vector. Impulse from a force has the SAME direction as the force. Tackling at an angle gives angular impulse — affects spin, not just translation.
• Confusing specific impulse with regular impulse. Specific impulse is a measure of rocket fuel efficiency (s). Regular impulse is total momentum change (N·s). Different quantities.