Ballistic Pendulum Calculator
Calculate the speed of a projectile from a ballistic pendulum experiment. Enter the projectile and pendulum masses plus the swing height or the swing angle and pendulum length.
Find the Projectile's Speed from a Ballistic Pendulum Experiment
Ballistic Pendulum Formula
Where:
- \(v\) — speed of the projectile immediately before impact (m/s)
- \(m\) — mass of the projectile (kg)
- \(M\) — mass of the pendulum block (kg)
- \(u = \sqrt{2gh}\) — speed of the combined projectile + block right after impact (m/s)
- \(g\) — acceleration due to gravity, 9.80665 m/s\(^2\)
- \(h\) — vertical rise of the block during the swing (m)
- \(L\) — pendulum length from pivot to centre of mass (m)
- \(\theta\) — maximum swing angle from the vertical (degrees)
The formula combines two physical laws: momentum is conserved during the perfectly inelastic collision, \(mv = (m + M)u\), and mechanical energy is conserved during the upward swing, \(\tfrac{1}{2}(m + M)u^2 = (m + M)gh\). Solving both together gives the projectile's speed before impact.
Ballistic Pendulum – Calculation Example
Example: a 10 g projectile embeds itself in a 250 g pendulum block. The block swings up to a maximum angle of 30\(\degree\) on a pendulum of length 0.5 m. What was the projectile's speed?
First find the vertical rise of the swing:
\[ h = L (1 - \cos\theta) = 0.5 \times (1 - \cos 30\degree) = 0.06699 \text{ m} \]
Then the speed of the combined mass just after impact:
\[ u = \sqrt{2 g h} = \sqrt{2 \times 9.80665 \times 0.06699} = 1.146 \text{ m/s} \]
Finally, apply conservation of momentum:
\[ v = \frac{m + M}{m} \, u = \frac{0.01 + 0.25}{0.01} \times 1.146 = 29.80 \text{ m/s} \]
Result: the projectile was travelling at about 29.80 m/s (107.3 km/h, 97.8 ft/s) before impact. The combined mass moved at only 1.146 m/s after impact, and 96.2% of the initial kinetic energy (about 4.27 J) was lost to heat and deformation in the collision.
How to Use the Calculator
- Enter the Projectile mass (m) and choose its unit (g, kg, lb or oz). This is the mass of the bullet, ball or pellet that lodges in the block.
- Enter the Pendulum mass (M) with its unit — the mass of the block alone, without the projectile.
- In How is the maximum swing measured?, choose whether you recorded the vertical rise directly (By swing height (h)) or measured the angle and pendulum length (By swing angle and length (\(\theta\), L)).
- Depending on your choice, enter either the Maximum swing height (h), or both the Maximum swing angle (\(\theta\)) in degrees and the Pendulum length (L) from the pivot to the centre of mass of the loaded block.
- Read the results: the projectile speed before impact in several units, the combined speed after impact, the swing height, and a breakdown of the energy lost in the collision.
What Is a Ballistic Pendulum?
A ballistic pendulum is a classic physics device for measuring the speed of a fast projectile that is difficult to time directly. The projectile is fired horizontally into a heavy block hung from long strings or rods. The projectile embeds itself in the block, the block swings upward, and the maximum rise — read off a scale or as a swing angle — tells you how fast the projectile must have been travelling.
The analysis works in two stages because two different conservation laws apply. During the very brief impact, kinetic energy is not conserved — much of it goes into deforming the block and the projectile — but momentum is. This gives the speed \(u\) of the combined mass right after impact. During the slow upward swing that follows, friction and air resistance are negligible, so mechanical energy is conserved: kinetic energy converts into gravitational potential energy, which is what the swing height measures.
Notice that the projectile speed is always larger than the post-impact speed by the mass ratio \((m + M)/m\). With a light projectile and a heavy block, this ratio is big — in the example above it is 26 — so a modest swing of the block corresponds to a fast projectile. This is also why the calculator reports a large energy loss: losing kinetic energy in the collision is not an error, it is the whole reason the pendulum measurement is possible without timing equipment.
Historically, ballistic pendulums were used from the 18th century onward to measure muzzle velocities of firearms, and the same principle still appears in undergraduate laboratories. It remains a clear demonstration that momentum and energy conservation each hold in their own domain, and that confusing the two leads to wrong answers.
Assumptions, Limitations and Practical Tips
- Perfectly inelastic collision. The projectile must embed in the block and stay there. If it bounces off or passes through, momentum transfer differs and the result will be wrong.
- Instantaneous impact. Gravity and string forces are assumed negligible during the collision, which is valid because the impact lasts only milliseconds.
- Frictionless swing. The pivot and air drag are neglected, so the measured swing height slightly underestimates the true post-impact speed. Real pendulums lose a few percent of their swing to friction.
- Pendulum length to the centre of mass. When using the angle method, \(L\) must be measured from the pivot to the centre of mass of the block with the projectile embedded, not the string length.
- Angles near 180\(\degree\). The height formula \(h = L(1 - \cos\theta)\) becomes very sensitive near 180\(\degree\); small angle-measurement errors then cause large speed errors. Small swings are also problematic because the height becomes hard to measure accurately — angles between roughly 10\(\degree\) and 60\(\degree\) give the best precision.
- Block much heavier than projectile. A large mass ratio amplifies measurement uncertainty in the masses; keep at least one decimal of precision on \(m\).
If you are studying projectile motion further, you can compare the computed speed with the speed of sound in air to see whether the projectile is supersonic, estimate how the launch angle and pendulum geometry interact through the Froude number, or account for velocity loss in flight with the drag force.
Frequently Asked Questions
Why can't I just set the projectile's kinetic energy equal to the potential energy at the top of the swing?
That is the most common mistake in this experiment. Kinetic energy is not conserved during the impact — typically over 90% of it turns into heat, deformation and sound as the projectile embeds in the block. Only momentum survives the collision unchanged. Equating \(\tfrac{1}{2}mv^2 = (m+M)gh\) instead of using \(v = \frac{m+M}{m}\sqrt{2gh}\) will underestimate the speed by a large factor.
What happens if the projectile bounces off or passes through the block?
The result will be wrong, because the formula assumes a perfectly inelastic collision where the projectile and block move together afterwards. If the projectile bounces back, momentum transfer is larger than assumed and your speed is underestimated; if it punches through, transfer is smaller and the speed is overestimated. Use a block material soft or thick enough to trap the projectile completely, such as clay, wax or wood of sufficient depth.
How do I find the centre of mass of the loaded pendulum for the length L?
Balance the block (with a projectile of the same type embedded in it) on a knife edge or your finger — the balance point is the centre of mass. Measure from the pivot to that point, not the full string or rod length. Alternatively, compute it: \(L = \frac{M L_{block} + m L_{projectile}}{m + M}\), where each length is measured from the pivot to that object's own centre. For a heavy block the difference from the block's centre is small but measurable.
Why does my calculated speed differ from a chronograph reading?
Three main reasons. First, real pendulums lose a few percent of their swing to pivot friction and air drag, so the height is slightly low and the speed underestimated. Second, the projectile loses some speed in flight between the muzzle and the block — a chronograph at the muzzle measures a different thing. Third, if the impact point is not in line with the block's centre of mass, the block rotates and wastes energy, again biasing the result low. Expect agreement within a few percent in a careful lab setup.
Does it matter if I enter masses in grams and lengths in inches or feet?
No — the calculator converts every unit to SI internally (kilograms and metres), so you can mix g, kg, lb and oz for the masses and m, cm, mm, ft or in for lengths and heights. The one input with no unit choice is the swing angle, which must always be in degrees from the vertical, not radians. Only the outputs are fixed in m/s (with km/h, ft/s and mph equivalents shown).
What swing angle and mass ratio give the most accurate results?
Aim for a maximum swing angle between about 10° and 60°. Below 10° the height is tiny and hard to measure; above about 60°, and especially near 180°, small angle errors blow up because \(1 - \cos\theta\) changes very steeply. Keep the block roughly 20–100 times heavier than the projectile: heavier blocks swing less (worse height precision), while lighter blocks make the mass ratio sensitive to weighing errors.
Can this calculator be used for arrows, airsoft BBs or paintballs?
Yes, as long as the projectile embeds in the block and stays there — arrows in a straw or foam target block are a classic variant. Airsoft BBs and paintballs often shatter or bounce, which breaks the perfectly inelastic assumption, so use a soft catching medium like modelling clay. The same two-stage physics applies: momentum conservation at impact, then energy conservation during the swing.