The Ballistic Pendulum: Two Conservation Laws, One Bullet Speed

A speed too fast to time directly

Before the age of chronographs and radar, measuring a bullet's muzzle velocity - typically hundreds of meters per second - was not something you could do by watching a stopwatch. The ballistic pendulum, developed by Benjamin Robins in the 1740s, sidesteps the problem entirely: fire the bullet into a large, softly padded wooden block hanging from a pivot, let the bullet embed itself, and measure something slow and easy instead - how high the whole block swings afterward. From that single swing height, two conservation laws in sequence hand you back the bullet's original speed.

Stage 1: the collision - momentum, not energy

A bullet of mass m traveling at speed v embeds in a stationary block of mass M. This is a perfectly inelastic collision - the two objects end up moving together as one - and the correct law to apply here is conservation of linear momentum, which holds for any collision regardless of whether energy is dissipated, because momentum conservation only requires the absence of external horizontal forces during the brief impact:

m v  =  (m + M) V          (momentum conserved through the impact)

V  =  m v / (m + M)         V = combined speed right after embedding

It is tempting, and wrong, to also apply energy conservation across this same impact. Kinetic energy is not conserved in a perfectly inelastic collision - a large fraction is converted into heat, sound and the plastic deformation that lets the bullet embed rather than bounce off. Applying ½mv² = ½(m+M)V² here would silently assume no energy was lost, and would return a bullet speed far lower than the truth.

Quantifying the loss

KE before  = ½ m v²
KE after   = ½ (m+M) V² = ½ (m+M) [m v/(m+M)]² = ½ m²v² / (m+M)

fraction of KE RETAINED  =  m / (m + M)

For a light bullet embedding in a much heavier block - a 10-gram bullet and a 2-kilogram block, say - the fraction retained works out to about 0.5%: over 99% of the bullet's original kinetic energy is gone, dissipated as heat and deformation in a fraction of a millisecond. That is a real, large, physically necessary loss, not a modeling inconvenience - and it is precisely why the second stage of the analysis has to use a completely different quantity.

Stage 2: the swing - now energy conservation applies

Once the bullet is embedded, the collision is over. From that instant onward, the combined bullet-and-block system simply swings on the pendulum's arm, converting its kinetic energy into gravitational potential energy as it rises, with no further collision and (idealizing away small friction and air resistance) no further energy loss. Here, energy conservation is exactly the right tool:

½ (m+M) V²  =  (m+M) g h        (KE right after impact → PE at peak swing)

V  =  √(2 g h)                   h = measured rise of the pendulum

combine with Stage 1 to solve for the original bullet speed v:
v  =  [(m+M)/m] · √(2 g h)

The height h at the top of the swing is easy to measure - with a simple ratchet catch marking the highest point reached, a ruler is all Robins needed. Feeding that h back through both equations recovers v, the bullet's speed at the moment of impact, entirely from static, low-speed measurements: the masses m and M, and the swing height h.

The general lesson: match the law to the physics

The ballistic pendulum is a small, self-contained lesson in when each conservation law applies. Momentum conservation is essentially unconditional for an isolated system during a collision, whether the collision is elastic or not. Energy conservation, on the other hand, needs a genuinely conservative process to hold - true for the swing (gravity only, no impact), false for the embedding impact itself (friction, deformation, heat). Mixing them up, or applying the wrong one to the wrong stage, is one of the most common errors in introductory mechanics, and this device makes the distinction concrete and measurable rather than abstract.

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