From Crime Labs to Classrooms

The ballistic pendulum is a classic experiment that blends collision physics with simple harmonic motion. In its tra ditional form a small projectile such as a bullet of mass $\mathrm{m_b}$ strikes and embeds in a larger wooden block or metal catcher of mass $\mathrm{m_p}$. The catcher is suspended as a pendulum and the impact causes t he combined system to swing upward to a height $h$. Before electronic chronographs existed, investigators and re searchers used the rise height to back–calculate the initial speed of the projectile, providing valuable forensic evidence or co ncepts in introductory physics laboratories.

The method relies on two fundamental conservation principles. Immediately after impact, the projectile and pendulum m ove together with some velocity $V$. The collision itself is perfectly inelastic because the bullet becomes lodge d in the catcher. Over the very brief interval of impact we assume external impulses such as tension in the supporting strings ar e negligible, so the total momentum of the bullet–pendulum system is conserved. This gives the first key relationship:

$$\mathrm{m_b}\mathrm{v_b}=\mathrm{(m_b}+\mathrm{m_p)}V$$

As the combined masses swing upward they gradually slow until coming momentarily to rest at the maximum height. During t his stage mechanical energy is conserved: kinetic energy at the bottom of the swing transforms into gravitational potential energ y at the top. That second principle yields

$$\frac{1}{2}\mathrm{(m_b}+\mathrm{m_p)}{V}^2=\mathrm{(m_b}+\mathrm{m_p)}gh$$

Simplifying the energy equation shows that the velocity right after the collision is $V=\sqrt{2gh}$. Substituting this into the momentum expression solves for the projectile speed

$$\mathrm{v_b}=\frac{\mathrm{(m_b}+\mathrm{m_p)}}{\mathrm{m_b}}\sqrt{2gh}$$

The expression illustrates the pendulum’s diagnostic power. A larger pendulum mass or higher rise height increases the r equired projectile speed, while a heavier projectile reduces the scaling factor. Because gravitational acceleration $gappears\; under\; a\; square\; root,\; variations\; in\; local\; gravity—say\; between\; Earth\; and\; the\; Moon—produce\; only\; modest\; changes\; in\; the\; in\; ferred\; speed,\; but\; our\; calculator\; allows$ g$to\; be\; adjusted\; for\; experiments\; on\; other\; worlds\; or\; precision\; laborator\; y\; setups.$

How to Use the Calculator

The input form above is flexible: it can compute any one of the variables provided the other four are supplied. To fi d the bullet speed after a test firing, enter the bullet mass, pendulum mass, measured rise height, and local gravitational accel between them. Leaving the bullet speed field blank will trigger the calculator to use the formula above. Conversely, if you kno w the speed from a chronograph test and wish to determine the resulting height, leave the height field empty and the program will solve for $h$. Solving for one of the masses is likewise possible, though it requires that the bullet speed be higher than the threshold $\sqrt{2gh}$ for physical consistency.

Internally the script examines which entry box is left blank. It then performs the corresponding algebraic rearrangem ent of the main relation. When solving for the projectile or pendulum mass, the code ensures the denominator does not vanish and provides a gentle warning if the combination of inputs would require a negative mass or imaginary velocity. In typical laborator y scenarios bullet masses range from a few grams to tens of grams while pendulum blocks may weigh several kilograms, so the ratio $\frac{\mathrm{m_b}}{\mathrm{m_p}}$ is quite small. This means the swing velocity $V$ is much smaller than the original bullet speed, highlighting how a massive catcher absorbs momentum.

A Historical Perspective

The ballistic pendulum was invented in the 1740s by English mathematician Benjamin Robins. His device revolutionized ballistics by providing a relatively simple way to measure the muzzle velocity of artillery, which at the time could not be time d accurately with mechanical clocks. Robins' pendulum consisted of a heavy wooden block suspended by long cords. A projectile wa s fired into the block, and the subsequent swing angle was measured using a graduated arc. By analyzing the motion he could inf er the projectile’s speed. The technique quickly spread across Europe, influencing military engineering and the emerging science of internal ballistics. Variants of the experiment appear in physics education to this day because it demonstrates how collisio n theory and energy conservation mesh together in one apparatus.

Modern forensic labs rarely use ballistic pendulums, having replaced them with electronic chronographs, Doppler radars , and high-speed cameras. Nevertheless the experiment retains historical charm and pedagogical value. It also surfaces in more o bscure settings such as testing the performance of air rifles or validating physics simulation software. The problem of a bullet embedding in a pendulum even finds analogies in astrophysics where colliding objects merge and then settle into new orbits under gravity's influence.

Worked Example

Suppose a 0.009 kg bullet embeds in a 3 kg block that swings up by 0.15 m. Setting $g=9.81$ m/s², the velocity immediately after impact is $V=\sqrt{2gh}=\sqrt{2\times 9.81\times 0.15}\approx 1.72$ m/s. Applying the momentum relation yields

$$\mathrm{v_b}=\frac{0.009+3}{0.009}\times 1.72\approx 575$$ m/s. The bullet’s actual initial speed is more than 300 times larger than the velocity of the recoiling block and roughly twice the speed of sound. Our calculator performs the same sequence of steps automatically and reports the value rounded to the desired precision.

Typical Projectile Masses

The table below lists representative bullet masses to provide context when entering values. Converting grams to kilogra ms ensures compatibility with the calculator’s SI units.

Caliber	Mass (g)
.22 LR	2.6
9 mm	7.5
.45 ACP	15.0
12 ga slug	28.3

Light rimfire rounds produce small pendulum deflections, while heavy shotgun slugs can move massive blocks. By adjustin g the masses in the form one can explore how the sensitivity of the apparatus changes and why precision experiments often use a large catcher relative to the projectile.

Assumptions and Limitations

Several idealizations underpin the ballistic pendulum model. Air resistance during the swing is ignored, although th e effect is usually minor over the short arc involved. The collision is treated as perfectly inelastic; if the projectile ricoche s or only partially embeds, the formula underestimates the bullet’s true velocity. The pendulum is assumed to start from rest a nd to swing in a vertical plane without twisting. Finally, the supporting cords must be rigid enough that their change in tension during impact does not contribute significant external impulse. Real experiments mitigate these issues through careful design, b ut users should be aware of the underlying simplifications when applying the calculator’s results to practical scenarios.

Extending the Analysis

The same conservation approach can analyze other collision devices. Replacing the pendulum with a cart on a track crea tes a linear ballistic trolley. Adding sensors to measure the velocity at several points yields insight into energy dissipation mechanisms. In engineering, similar equations estimate the recoil of firearms and the performance of ballistic shields. In each case, the combination of momentum conservation during a short impulse and energy conservation over longer motion provides a robu st framework for understanding dynamics.

Whether for historical curiosity or classroom exploration, the ballistic pendulum remains a tangible bridge between c oncepts of momentum, energy, and gravitational potential. This calculator encapsulates the essential mathematics in an easy-to-us e interface, freeing students and hobbyists to focus on the fascinating physics that unfolds during the brief moment of impact an d the graceful subsequent swing.