Ballistic pendulum

A projectile embeds in a hanging block, then the combined mass swings upward. The collision is inelastic, so momentum is the useful law during impact. After impact, mechanical energy connects the rise height to the speed just after collision.

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Speed just after impact

V₀ = mv / (M + m)

Maximum angle

cos θ_max = 1 - V₀² / (2gL)

Rise of the center

h = L(1 - cos θ_max)

Inferred launch speed

v = (M + m) / m · √(2gh)

Momentum check

mv = (M + m)V₀

Current bob speed

|v| = L|ω|

Projectile mass m

Pendulum mass M

Launch speed v

Pendulum length L

Playback speed

show velocity vectors

What to try. Set v very small and confirm the rise is tiny. Keep v fixed and increase M: the post-collision speed and the peak angle both drop. Keep M fixed and increase m: more momentum goes into the block and the peak angle rises.

Units: kilograms, meters, seconds; g = 9.8 m/s². Assumptions: the projectile is launched horizontally at the bob height with pre-impact drop neglected, collision is perfectly inelastic, the string is massless and rigid, the pivot is frictionless, and air drag is ignored. The run stops at the first peak angle.