Ballistic Pendulum Calculator

A ballistic pendulum is a device used to measure the velocity of a projectile such as a bullet. The projectile embeds itself into a hanging block (the pendulum), and the combined system swings upward. By measuring the height of the swing and applying conservation of momentum (during the collision) and conservation of energy (during the swing), the original projectile velocity can be calculated.

The collision between the projectile and the pendulum block is a perfectly inelastic collision — the two objects stick together. This type of collision is inherently dissipative: kinetic energy is converted into heat, sound, and deformation of materials. Conservation of energy only applies to the swing phase after the collision, where the kinetic energy of the combined mass converts into gravitational potential energy.

The calculation combines two conservation laws. First, conservation of momentum: mv = (m + M)V, where m is projectile mass, M is pendulum mass, v is initial projectile velocity, and V is the post-collision velocity. Second, conservation of energy during the swing: ½(m+M)V² = (m+M)gh, giving V = √(2gh). Combining these: v = ((m + M) / m) × √(2gh). If an energy loss factor η is included, v = ((m + M) / m) × √(2gh / η).

The energy loss factor (η) accounts for real-world inefficiencies in the pendulum swing, such as air resistance, friction at the pivot, and elasticity losses. A value of 1.0 represents a perfectly ideal pendulum (no losses), while values less than 1 reduce the effective height used in the calculation, yielding a higher (more realistic) projectile velocity estimate.

All masses should be entered in kilograms (kg), height in meters (m), and gravitational acceleration in m/s². The calculator returns velocity in meters per second (m/s), momentum in kg·m/s, and energy values in joules (J). Ensure consistency — mixing units (e.g., grams with meters) will produce incorrect results.

In a typical ballistic pendulum experiment, a very large percentage of kinetic energy is lost — often 95% or more. This is because the bullet's mass is tiny compared to the block, so most of the energy is converted to heat, deformation, and sound during the inelastic collision. Only a small fraction transfers into the swing of the combined system.

Yes. The ballistic pendulum principle applies to any projectile that embeds in or sticks to a hanging mass — including arrows, balls, or other objects. As long as the collision is perfectly inelastic (the projectile stays lodged in the pendulum), the same conservation laws apply. Simply enter the appropriate masses, rise height, and gravitational acceleration for your setup.

The post-collision velocity V is the speed at which the combined projectile-and-block system begins to swing immediately after impact. It is always much smaller than the original bullet velocity because the large mass of the block absorbs most of the momentum. V = √(2gh) and is used as the bridge between the momentum phase (collision) and the energy phase (swing).