Ballistic pendulum

A projectile of mass mᵦ strikes the stationary pendulum mass mᴮ horizontally at the bottom and sticks to it. The ideal support is a massless rigid rod of length L with point masses at its end. It can push as well as pull. This is not a string pendulum or a massive physical rod; friction, air resistance and pivot losses are excluded. Gravity is g = 9.81 m/s².

During the instantaneous impact, horizontal momentum is conserved: mᵦv = (mᵦ + mᴮ)V. The before/after kinetic energies are ½mᵦv² and ½(mᵦ + mᴮ)V². Their difference is energy lost in the inelastic impact. For nonzero impact speed, the lost fraction is mᴮ/(mᵦ + mᴮ). When v = 0, the displayed loss fraction is defined as zero, since both energies are zero.

After impact, the combined mass moves without damping: θ̇ = ω and ω̇ = −(g/L)sinθ. Height is h = L(1 − cosθ), kinetic energy K = ½(mᵦ + mᴮ)L²ω² and potential energy U = (mᵦ + mᴮ)gh. K + U remains constant; linear momentum does not remain constant during the subsequent swing.

The energy-equivalent height is hₑ = V²/(2g). For 0 < hₑ < 2L, the pendulum turns at θₘₐₓ = acos(1 − hₑ/L). At hₑ = 2L, the separatrix approaches 180° and height 2L only asymptotically. Above this threshold, the rigid-rod pendulum rotates and reaches height 2L with nonzero speed; no turning angle is reported. Zero speed gives rest. Classification at the threshold allows only a small floating-point tolerance.

Run starts with an approximately 0.4-second schematic approach, independent of the selected playback speed. It is not projectile flight time; swing time begins at impact. Step skips the approach and advances swing time by 0.05 s. Pause, input changes and leaving the tool stop playback. Reset clears the result while retaining inputs. Valid inputs are retained when navigating back. The drawing fits the full circular path for every rod length; displayed angles are measured from the downward vertical and accumulate through full turns.

Oscillations and rotations use the shared fourth-order Runge–Kutta integrator with small bounded substeps. The separatrix uses its analytical solution to avoid numerical drift into rotation. Sources: UCSB: ballistic pendulum and MIT: pendulum dynamics.