Why is the collision assumed to be perfectly inelastic? Doesn't that mean energy is lost?

Yes, the collision is perfectly inelastic because the bullet embeds in the block and they move together. This maximizes the loss of kinetic energy during the impact stage, converting it mostly into heat and sound. This energy loss is why we cannot use energy conservation for the collision itself. We must use momentum conservation, which holds true for the isolated system during the brief impact, regardless of energy loss.

Is the small-angle approximation necessary? What if the swing is large?

The small-angle approximation (sin θ ≈ θ) simplifies textbook algebra for tiny amplitudes (roughly θ ≪ 1 rad), linking ω₀ = V/L to θ_max only in that limit. For larger swings it fails (often noticeably past about 15–20°). This simulator does not use that approximation for the reported θ_max or the motion: the readout comes from exact energy (1/2)(m+M)V² = (m+M)gL(1 - cos θ_max), and the trajectory uses θ'' = -(g/L) sin θ, which remains valid at any amplitude up to the ideal-model assumptions.

How is this used in real-world forensics or ballistics?

Historically, ballistic pendulums were used to measure muzzle velocities of firearms before modern electronics. By firing a bullet into a heavy, suspended block and measuring the swing, the initial bullet speed could be calculated using the principles in this simulator. While obsolete for high-precision work, it remains a fundamental lab experiment for teaching conservation laws.

What are the main limitations of this idealized model?

The model ignores several real-world factors: air resistance during the swing, friction at the pivot point, the rotational inertia of the pendulum bob (treated as a point mass), and any elasticity in the collision or the pendulum rod. These simplifications allow us to focus on the core physics, but in a real experiment, they would introduce systematic errors that require careful setup to minimize.

Ballistic Pendulum

Bullet hits block: embedded vs e; ω₀ = v/L, θ_max, energy graph.

Live graphs

About this model

A ballistic pendulum provides a classic method for determining the velocity of a projectile without direct high-speed measurement. This simulator models the two-stage process: a perfectly inelastic collision where a bullet embeds itself in a wooden block, followed by the conservation of mechanical energy as the combined block-bullet system swings upward. The first stage is governed by the conservation of linear momentum. For a bullet of mass m and initial velocity v_bullet striking a stationary block of mass M, the post-collision velocity V of the combined mass is given by V = (m * v_bullet) / (m + M). The second stage converts kinetic energy into gravitational potential. With no drag or pivot friction, the displayed maximum angle follows from (1/2)(m+M)V² = (m+M)gL(1 - cos θ_max), and the animated swing integrates the full nonlinear pendulum equation θ'' = -(g/L) sin θ (velocity Verlet), not the small-angle linearization sin θ ≈ θ. Textbooks often introduce that small-angle limit, where ω₀ = V/L and ω₀² ≈ (g/L) θ_max for tiny swings only. The simulator also offers an optional 1D impact with restitution e, where only the block mass M swings. Plots show kinetic, potential, and total energy versus time. Key simplifications include a massless, rigid rod; negligible air resistance and friction at the pivot; and an instantaneous collision. By interacting with this model, students can explore momentum conservation in isolated systems, energy transfer in conservative motion, and combining these ideas to infer an unknown initial speed.

Who it's for: High school and introductory university physics students studying the conservation laws of momentum and energy, particularly in the context of inelastic collisions and pendulum motion.

Key terms

Conservation of Momentum
Conservation of Energy
Perfectly Inelastic Collision
Ballistic Pendulum
Angular Velocity
Gravitational Potential Energy
Small Angle Approximation
Projectile Velocity

How it works

A bullet strikes a hanging block at the lowest point of the arc so the impulse is approximately horizontal along the pendulum’s tangent. In the classic fully inelastic case the bullet embeds, momentum gives (M+m)V = mv₀, and the subsequent swing conserves mechanical energy in this ideal model (no drag). With a nonzero restitution e we treat a one-dimensional impact along the same line: the block picks up v₂′ = (1+e)mv₀/(m+M) while the bullet rebounds with v₁′ = v₀ − (1+e)Mv₀/(m+M); only the block is attached to the string, so its initial angular speed is v₂′/L. Compare predicted maximum angle from ½Mv₂′² = MgL(1−cos θ_max) with the live simulation.

Key equations

Embedded: (M+m)V = mv₀ · ½(M+m)V² = (M+m)gL(1−cos θ_max)

General: v₂′ = (1+e)mv₀/(m+M) · ω₀ = v₂′/L

Frequently asked questions

Why is the collision assumed to be perfectly inelastic? Doesn't that mean energy is lost?: Yes, the collision is perfectly inelastic because the bullet embeds in the block and they move together. This maximizes the loss of kinetic energy during the impact stage, converting it mostly into heat and sound. This energy loss is why we cannot use energy conservation for the collision itself. We must use momentum conservation, which holds true for the isolated system during the brief impact, regardless of energy loss.
Is the small-angle approximation necessary? What if the swing is large?: The small-angle approximation (sin θ ≈ θ) simplifies textbook algebra for tiny amplitudes (roughly θ ≪ 1 rad), linking ω₀ = V/L to θ_max only in that limit. For larger swings it fails (often noticeably past about 15–20°). This simulator does not use that approximation for the reported θ_max or the motion: the readout comes from exact energy (1/2)(m+M)V² = (m+M)gL(1 - cos θ_max), and the trajectory uses θ'' = -(g/L) sin θ, which remains valid at any amplitude up to the ideal-model assumptions.
How is this used in real-world forensics or ballistics?: Historically, ballistic pendulums were used to measure muzzle velocities of firearms before modern electronics. By firing a bullet into a heavy, suspended block and measuring the swing, the initial bullet speed could be calculated using the principles in this simulator. While obsolete for high-precision work, it remains a fundamental lab experiment for teaching conservation laws.
What are the main limitations of this idealized model?: The model ignores several real-world factors: air resistance during the swing, friction at the pivot point, the rotational inertia of the pendulum bob (treated as a point mass), and any elasticity in the collision or the pendulum rod. These simplifications allow us to focus on the core physics, but in a real experiment, they would introduce systematic errors that require careful setup to minimize.

Continue with similar topics in this category — or all 85 in Classical Mechanics.

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