Visualise the classic ballistic pendulum — fire a bullet into a hanging block and recover the bullet's speed from how high and how far the block swings. Adjust the bullet mass and speed, the block mass and the pendulum length to see the two stages clearly: momentum conservation in the inelastic collision and energy conservation in the swing-up.
Parameters
Bullet mass m
g
Mass of the bullet that embeds in the block
Pendulum (block) mass M
kg
Mass of the hanging target block
Bullet speed v_b
m/s
Muzzle speed of the bullet just before impact
Pendulum length L
m
Length from the pivot to the block's centre
Results
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Post-impact velocity (m/s)
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Swing height (mm)
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Swing angle θ (deg)
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Kinetic energy before impact (J)
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Energy lost in collision (J)
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Energy-loss fraction (%)
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Ballistic pendulum animation
The bullet is fired horizontally into the block; the combined mass swings up to the angle θ and back. Stage 1 is momentum conservation, stage 2 is energy conservation.
Common velocity v just after impact (momentum conservation) and swing height h (energy conservation). m: bullet mass, M: block mass, v_b: bullet speed, g: gravitational acceleration.
$$\theta=\arccos\!\left(1-\frac{h}{L}\right)$$
Swing angle θ. L: pendulum length. The angle the pendulum swings through follows from h and L.
$$\frac{\Delta E}{E_b}=\frac{M}{m+M}$$
Fraction of kinetic energy lost in the inelastic collision. Momentum is conserved in the (inelastic) collision and mechanical energy in the swing-up — the two stages must not be mixed.
What is the Ballistic Pendulum Simulator?
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A "ballistic pendulum" is that experiment where you fire a bullet into a block of wood, right? What is it actually for?
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Exactly. Its purpose is simply to measure how fast the bullet is going. A bullet is far too fast to time with a stopwatch. So you fire it into a heavy block hanging on cords; the bullet buries itself, and the block swings up slowly. That swing height you can measure with a ruler. Before electronic chronographs existed, this was the standard instrument for finding a bullet's muzzle speed. It was devised back in 1742 by the English mathematician Benjamin Robins.
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I see! But how do you get the bullet's speed out of the swing height?
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This is the most interesting part of the ballistic pendulum: you have to split the event into two distinct stages. The first stage is the instant of the collision, as the bullet buries itself in the block. That is a perfectly inelastic collision, and momentum is conserved through it. The second stage is the swing-up of the now-combined block. There is no collision there, only gravity, so mechanical energy is conserved. Apply those two in order and you can work backwards from the swing height to the bullet's speed.
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Wait — momentum is conserved in the collision, so isn't energy conserved too?
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That is the biggest pitfall, and the answer is no. At the instant of impact momentum is conserved, but kinetic energy absolutely is not. As the bullet ploughs into the block, most of the kinetic energy turns into heat and deformation of the block. Leave the sliders at their defaults and look at the results — the energy-loss fraction is over 99%. The fraction lost is exactly M/(m+M); since the bullet is hugely lighter than the block, that is almost 1 — almost a total loss.
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Over 99% lost, yet momentum survives intact... so why is energy conserved in the second stage, the swing-up?
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Because there is no "collision" in the second stage. Once the bullet is embedded and the two move as one, there is no more sudden, violent interaction. The pendulum simply rises under gravity. If we ignore friction and air drag, the kinetic energy it had right after impact, ½(m+M)v², converts cleanly into potential energy at the top, (m+M)gh. That is why the simple relation h = v²/(2g) holds. Energy is lost only during the collision — that is the rule to remember.
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In a real measurement, how do you go from the swing height back to the bullet speed?
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You trace it backwards. First measure the swing height h (or the angle θ). Energy conservation, h = v²/(2g), gives you the common velocity v just after impact. Then put that v into momentum conservation, v = m·v_b/(m+M), and solve for the bullet speed v_b. Use energy conservation to find the post-impact speed, then momentum conservation to find the bullet speed — the order matters. Reverse it and the answer falls apart. Try moving the speed and mass sliders here to feel how the two stages link together.
Frequently Asked Questions
Because the event splits into two stages with very different physics. The first stage is the perfectly inelastic collision as the bullet buries itself in the block: the impulse of external forces is negligible, so momentum is conserved, but kinetic energy is largely lost to heat and deformation. The second stage is the swing-up of the block with the embedded bullet: there is no collision and only gravity acts, so mechanical energy is conserved. Even though both are conservation laws, the conserved quantity differs, so you must apply the correct law in each stage. Blurring them into a single equation gives a badly wrong bullet speed.
First, from momentum conservation, the common velocity of the bullet (mass m, speed v_b) and block (mass M) immediately after they lock together is v = m·v_b /(m+M). Next, from energy conservation, treating that kinetic energy as entirely converted to potential energy, the swing height is h = v²/(2g), where g is the gravitational acceleration (9.81 m/s²). The swing angle follows from the pendulum length L as θ = arccos(1 − h/L). This tool computes all of these automatically and also shows the kinetic energy before and after impact and the energy-loss fraction.
In a perfectly inelastic collision the fraction of kinetic energy lost is exactly M/(m+M). Because the bullet is far lighter than the block, this value is very close to 1: in a real ballistic pendulum more than 99% of the bullet's kinetic energy disappears at the instant of impact. For a 10 g bullet and a 5 kg block, for example, the loss fraction is 5.0/5.01 ≈ 99.8%. That large energy goes into heat and deformation of the block. Momentum, however, is not lost, so the small remaining kinetic energy swings the pendulum up.
Long before electronic chronographs, the ballistic pendulum was the standard laboratory instrument for measuring the muzzle speed of a bullet. Devised in 1742 by the English mathematician Benjamin Robins, its breakthrough was converting a speed too fast to time directly into a slowly measurable quantity — a swing. Because the apparatus is simple and it is an excellent teaching device for the difference between momentum and energy conservation, it is still widely used in university and high-school physics labs. It is one of the founding instruments of the science of ballistics.
Real-World Applications
Measuring projectile speed and ballistics: Until electronic chronographs became common, the ballistic pendulum was the de facto standard instrument for measuring a bullet's muzzle speed. From Benjamin Robins' 1742 invention through the 19th century it was used to evaluate firearm performance and improve gunpowder, and it underpinned the growth of ballistics as a discipline. The idea of converting an unmeasurably fast speed into a slowly measurable swing is a universal one that still echoes in modern measurement engineering.
A staple of physics education: The ballistic pendulum is one of the most widely used instruments in university and high-school physics labs. A spring launcher fires a small metal ball into a block, and the swing angle is read off to recover the launch speed. Because it makes you decide "when, and which" conservation law to use, it is regarded as one of the best teaching tools for the difference between elastic and inelastic collisions.
Collision analysis and safety design: Crash tests of cars and the evaluation of protective gear use the same "perfectly inelastic collision" framework as the ballistic pendulum. When two crashed vehicles move on together, or a bulletproof vest stops a round, momentum is conserved while a great deal of kinetic energy is absorbed. That absorbed energy is itself the measure of safety performance, and the ballistic pendulum is its simplest model.
Verifying CAE and numerical simulations: In explicit-FEM impact analysis (crash and penetration simulations), solvers are first validated on simple problems with analytical solutions, like the ballistic pendulum. Checking that momentum is conserved exactly and that the energy loss is physically reasonable catches mistakes in contact settings or time-step size early. The exact solution this tool provides can serve as the reference value for such sanity checks.
Common Misconceptions and Pitfalls
The biggest misconception is computing everything from collision to swing-up with a single energy-conservation equation. Assuming "the bullet's kinetic energy ½m·v_b² becomes the potential energy (m+M)gh directly" badly underestimates the bullet speed. The collision is perfectly inelastic, and over 99% of the kinetic energy vanishes into heat and deformation, so only a tiny remainder powers the swing-up. Correctly, you apply momentum conservation to the collision stage and energy conservation to the swing-up stage, in order. This is a decisive point for understanding the difference between elastic and inelastic collisions.
Next, the assumption that "if momentum is conserved, energy must be too". Momentum and energy are different quantities with different conditions for conservation. Momentum is conserved whenever the impulse of external forces is zero (or negligible), regardless of whether a collision is elastic or inelastic. Kinetic energy, however, is conserved only in elastic collisions. In the ballistic pendulum, a perfectly inelastic collision, momentum is conserved but kinetic energy is not — grasping this asymmetry is the key to solving collision problems.
Finally, treating the pendulum length L and the small-angle assumption lightly. This tool computes the swing height exactly as h = v²/(2g) and the swing angle as θ = arccos(1 − h/L). When you measure an angle experimentally to back out the speed, you use h = L(1 − cosθ); but for large swing angles the small-angle approximation sinθ ≈ θ no longer holds. A real pendulum is a rigid body, not a point mass, so a precise treatment models it as a physical pendulum with rotational inertia (moment of inertia). The point-mass approximation is fine for teaching, but for precision measurement, pivot friction, air drag and the block's shape are also sources of error.
How to Use
Enter bullet mass (bmNum, 0.01–0.05 kg range) and pendulum mass (pmNum, 0.5–5 kg range) using the sliders.
Set bullet velocity (vbNum, 100–500 m/s) and pendulum length (plNum, 0.5–2 m) before firing.
Click fire to embed the bullet; read post-impact velocity, swing height, angle θ, and energy metrics instantly.
Worked Example
Bullet mass 0.02 kg at 300 m/s embeds in pendulum (2 kg, 1 m length). Pre-impact kinetic energy: 0.5 × 0.02 × 300² = 900 J. Post-collision velocity via momentum conservation: (0.02 × 300)/(0.02 + 2) = 2.97 m/s. Swing height: v²/(2g) = 8.82/(19.62) = 0.45 m (450 mm). Energy lost: 900 − [0.5 × 2.02 × 2.97²] = 891.1 J, representing 99% inelastic loss.
Practical Notes
Steel bullets (7.62 mm NATO, ~9.7 g) into wood blocks demonstrate extreme momentum transfer; energy loss exceeds 98% in typical ballistic gelatin tests.