Ballistic Pendulum

The two-phase idea

The whole point of the ballistic pendulum is that a flying bullet is far too fast to time directly with seventeenth- or eighteenth-century instruments — a musket ball travels several hundred metres per second and crosses a measuring gap in a fraction of a millisecond. Robins’ insight was to trade speed for height: let the bullet bury itself in a heavy block, and measure the slow, leisurely swing that follows instead.

The event has two physically distinct phases, and the most common mistake is to blur them together:

• Phase 1 — the collision. The bullet embeds in the block. This is a perfectly inelastic collision: the two masses move off together. Momentum is conserved because the enormous impact forces are internal to the bullet-plus-block system and act over a vanishingly short time. Kinetic energy is not conserved — most of it becomes heat, sound, and crushed wood.
• Phase 2 — the swing. The combined block-plus-bullet, now moving slowly, swings up like a pendulum. Here there is no longer any sudden impact, so mechanical energy is conserved: the kinetic energy just after impact turns entirely into gravitational potential energy at the top of the swing.

Deriving the speed formula

Let m be the bullet mass, v its unknown speed, M the block mass, and V the speed of the combined mass immediately after impact. Phase 1 (momentum conservation):

m·v = (m + M)·V        →        V = m·v / (m + M)

Phase 2 (energy conservation during the swing, where the block rises by height h):

½(m + M)·V² = (m + M)·g·h        →        V = √(2gh)

Notice the combined mass cancels in the energy equation — the swing height tells you V regardless of how heavy the block is. Eliminate V between the two results:

v = (m + M)/m · √(2gh)

If you measure the maximum swing angle θ of a pendulum of length L instead of the height directly, substitute the geometry h = L(1 − cos θ):

v = (m + M)/m · √(2gL(1 − cos θ))

Why energy conservation fails across the collision

Suppose you naively assumed kinetic energy was conserved during the impact. You would set ½m·v² equal to (m + M)·g·h and solve for v. The result is wildly wrong because almost all the bullet’s energy is lost in the collision. The fraction of kinetic energy that survives is exactly the mass ratio:

KE_after / KE_before = m / (m + M)

For a typical 10 g bullet in a 2.0 kg block, that is 0.010 / 2.010 ≈ 0.5%. Over 99.5% of the kinetic energy is gone — vaporized into heat in the splintering wood — yet 100% of the momentum carries through. Momentum and energy obey different bookkeeping, and the ballistic pendulum is the cleanest illustration of why.

Quantity	Phase 1 — collision	Phase 2 — swing
Momentum	Conserved (internal forces)	Not conserved (gravity is external)
Kinetic energy	Mostly lost to heat / deformation	Conserved (converts to potential)
What you compute	Combined speed V from m·v	Speed V from height √(2gh)
Time scale	~1 millisecond	~1 second
Tool used	Conservation of momentum	Conservation of energy

A worked example

A m = 10 g bullet is fired into a M = 2.0 kg wooden block hanging on a L = 1.5 m cord. The block-plus-bullet swings up to a maximum angle of θ = 22°. Find the bullet speed.

h = L(1 − cos θ) = 1.5 × (1 − cos 22°) = 1.5 × (1 − 0.927) = 0.109 m
V = √(2gh) = √(2 × 9.81 × 0.109) = 1.46 m/s
v = (m + M)/m × V = (2.010 / 0.010) × 1.46 = 201 × 1.46 ≈ 294 m/s

So a swing of barely 11 cm reveals a bullet travelling at nearly the speed of sound. The leverage in the mass ratio (201× here) is what makes the device so sensitive: a tiny, easily measured displacement is amplified back into a large velocity.

Bullet (m)	Block (M)	Swing height h	Recovered speed v
10 g	2.0 kg	0.109 m	≈ 294 m/s (musket / pistol)
8 g	4.0 kg	0.041 m	≈ 450 m/s (rifle round, slowed)
4 g	1.5 kg	0.020 m	≈ 235 m/s (small calibre)
26 g	10 kg	0.080 m	≈ 482 m/s (cannon-scale, scaled block)

Computing it in code

const g = 9.81;

// Bullet speed from swing height (the classic ballistic-pendulum result)
function bulletSpeed(m, M, h) {
  const V = Math.sqrt(2 * g * h);        // speed just after impact (energy)
  return (m + M) / m * V;                // pre-impact bullet speed (momentum)
}

// If you measured the swing ANGLE instead of the height
function bulletSpeedFromAngle(m, M, L, thetaDeg) {
  const h = L * (1 - Math.cos(thetaDeg * Math.PI / 180));
  return bulletSpeed(m, M, h);
}

// Fraction of the bullet's kinetic energy that survives the collision
function energyRetained(m, M) {
  return m / (m + M);
}

const m = 0.010, M = 2.0, L = 1.5;
console.log(bulletSpeedFromAngle(m, M, L, 22).toFixed(0) + ' m/s');   // 294 m/s
console.log((energyRetained(m, M) * 100).toFixed(2) + '% KE kept');   // 0.50% KE kept

// Forward check: predict the swing height for a known 320 m/s bullet
function swingHeight(m, M, v) {
  const V = m * v / (m + M);             // momentum gives combined speed
  return V * V / (2 * g);                // energy gives the rise
}
console.log((swingHeight(0.010, 2.0, 320) * 1000).toFixed(1) + ' mm rise'); // ~129 mm

From point mass to real pendulum

The clean formula treats the block as a point mass swinging on a massless string. A real ballistic pendulum is a rigid body, so the honest analysis uses angular momentum about the pivot and the block’s moment of inertia I. Conservation of angular momentum at impact gives m·v·d = I·ω, where d is the impact height below the pivot and ω the angular speed just after. Energy conservation then raises the centre of mass by h_cm. For a compact, heavy block on a long cord the point-mass formula is accurate to a few percent; for a stubby pendulum the rotational correction matters and experimenters either compute I directly or calibrate the device with a projectile of known speed.

Where the ballistic pendulum shows up

• Historical ballistics. From Robins (1742) through Hutton, it set muzzle-velocity standards for muskets and cannon and seeded the entire science of exterior ballistics.
• Physics teaching. It is the canonical lab for distinguishing momentum conservation from energy conservation — arguably its most important modern role.
• Forensics and proof testing. Variants measure the momentum of fragments, airgun pellets, and arrows where electronic chronographs are awkward.
• Impulse measurement. Ballistic-pendulum-style rigs gauge the impulse of small rocket motors and explosive squibs.
• Sports and recreation. Paintball, archery, and slingshot enthusiasts build simple pendulum rigs to compare projectile energies.

Common mistakes

• Applying energy conservation to the collision. The single biggest error. The impact is inelastic; you must use momentum there and energy only during the swing.
• Forgetting the bullet mass in the swing. After impact the moving mass is m + M, not just M. For a heavy block the difference is tiny, but keep it in the algebra.
• Mixing up h and L. The rise is h = L(1 − cos θ), not L·θ or L·sin θ. Small-angle shortcuts introduce real error past about 15°.
• Ignoring the rotational correction. Treating a short, fat block as a point mass overestimates the speed. Use moment of inertia for a physical pendulum.
• Letting the bullet pass through. If the block is too light or thin the bullet exits and momentum bookkeeping breaks — the device only works for a clean, perfectly inelastic capture.