Ballistic pendulum simulator demonstrating the classic two-stage method for measuring projectile speed. Stage one: a bullet of mass m travelling at speed v embeds in a stationary pendulum bob of mass M in a perfectly inelastic collision, conserving momentum, m·v = (m+M)·V. Stage two: the combined mass swings upward, conserving mechanical energy, ½(m+M)V² = (m+M)gh, so the swing height h (or maximum angle) can be used to back-calculate the original bullet speed, v = ((m+M)/m)·√(2gh). The simulator also reports the large kinetic-energy loss in the collision itself, which does not violate overall energy conservation.
The ballistic pendulum solves a problem that plagued gunnery for two centuries: how do you measure the speed of something too fast to time by eye? The trick is to break the event into two stages that each obey a different conservation law. In stage one, a bullet of mass m moving at speed v embeds itself in a stationary wooden bob of mass M in a collision that is over in a fraction of a millisecond — far too fast for gravity to do any meaningful work. Because no external horizontal force acts during that instant, momentum is conserved: m·v = (m+M)·V, where V is the velocity of the combined bob-plus-bullet immediately afterward. This collision is perfectly inelastic — the bullet stays embedded — so kinetic energy is emphatically not conserved here; a large fraction is lost to deformation, heat and sound as the bullet buries itself in the wood.
In stage two, the surviving kinetic energy of the combined mass carries it upward along the pendulum's swing, and from the moment right after impact onward, only gravity and string tension act, and tension does no work. Mechanical energy is now conserved: ½(m+M)V² = (m+M)gh, so the bob rises to a measurable height h, or equivalently a maximum angle θmax = arccos(1 − h/L) for a pendulum of length L. Because every step is invertible, a measured swing height lets you work backward to the bullet's original speed: v = ((m+M)/m)·√(2gh). This is exactly the device Benjamin Robins built in 1742, decades before anyone had an electronic way to time a bullet — by swapping an unmeasurable speed for an easily measured swing, he turned interior and exterior ballistics into a quantitative science. The huge energy loss in the collision is not a violation of energy conservation; it is energy conservation working correctly across a system boundary, with the "missing" energy fully accounted for as heat and deformation rather than vanishing.
What does this simulation actually show?
It animates a bullet flying in and embedding in a stationary pendulum bob, then swinging up along a circular arc to a maximum angle before swinging back. The telemetry panel reports momentum just before and after the collision (equal, demonstrating conservation), the kinetic energy before and after the collision (very different, showing the loss), and the swing height and angle used to back-calculate the bullet's original speed.
Why does the collision lose so much energy but momentum is still conserved?
Momentum and kinetic energy are conserved under different conditions. Momentum is conserved in any collision with no net external horizontal force, regardless of how the collision deforms the objects. Kinetic energy is only conserved in perfectly elastic collisions. A bullet embedding in wood is the textbook example of a perfectly inelastic collision — the objects stick together — so momentum survives intact while most of the kinetic energy is converted to heat, sound and permanent deformation.
Who invented the ballistic pendulum and why?
The English mathematician Benjamin Robins introduced the ballistic pendulum in 1742, described in his book "New Principles of Gunnery." Before electronic chronographs existed, there was no direct way to time a musket ball travelling several hundred metres per second. Robins realised that if the ball's momentum could be transferred to a heavy, slow-moving pendulum, the resulting swing — easily measured with a simple angle scale — could be used to reconstruct the ball's original speed, transforming ballistics from guesswork into measurement.
The bullet mass slider (5–100 g) and bullet speed slider (50–500 m/s) set the incoming projectile. The bob mass slider (0.5–5 kg) sets the mass of the stationary pendulum it strikes, and the pendulum length slider (0.5–2 m) sets the length of the string or rod. The Fire button launches the two-stage sequence: collision, then swing.
Step one applies conservation of momentum to the instantaneous, perfectly inelastic collision: m·v = (m+M)·V, giving the post-collision speed V = m·v/(m+M). Step two applies conservation of mechanical energy to the swing that follows: ½(m+M)V² = (m+M)gh, giving the rise height h = V²/(2g), which converts to a maximum swing angle via h = L(1 − cos θmax).
Reading the maximum height (or angle) the bob reaches lets you reverse both steps. First recover the post-collision speed from the height, V = √(2gh); then reverse the momentum equation using the mass ratio, v = ((m+M)/m)·V = ((m+M)/m)·√(2gh). The simulator displays this "v measured" value alongside your input speed so you can confirm they match.
During the swing, the only forces doing work are gravity (conservative) and string tension, which acts perpendicular to the bob's motion and therefore does zero work — so mechanical energy is conserved throughout the swing. During the collision itself, the bullet and bob exert large internal contact forces on each other as the bullet decelerates and deforms the wood; those forces do net negative work on the pair's kinetic energy even though they cancel out in the momentum sum, which is why energy drops sharply at that single instant but not afterward.
As the bullet's mass m becomes small relative to the bob mass M, the post-collision speed V ≈ (m/M)v becomes tiny, and the fraction of kinetic energy that survives, KEafter/KEbefore ≈ m/M, becomes vanishingly small. This is why real ballistic pendulums use heavy bobs: a heavy bob barely moves, but that small motion is precisely measurable and losing over 99% of the kinetic energy in the collision is expected, not a flaw.
The core physics — conservation of momentum in the collision and conservation of mechanical energy in the swing — is exact and is exactly what a real ballistic pendulum relies on. The simulation omits secondary real-world effects such as air resistance on the swing, friction at the pivot, the finite size and moment of inertia of the bob (treated here as a point mass), and any energy radiated as sound, all of which introduce small corrections in a physical apparatus but do not change the underlying two-stage method.