---
title: "Ballistic Pendulum Lab Interactive Lab | Sandbox Physics"
description: "Can you work out a projectile's speed from how high the pendulum swings? Yes, if you work backward in two steps. The swing after the hit conserves energy, so the height the pendulum rises gives its speed just after impact. The hit itself loses energy but conserves momentum, and momentum turns that speed into the projectile speed. If the pendulum swings right over the top, its rise no longer tells you the speed."
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language: "en"
page_type: "lab"
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last_modified: "2026-10-22"
---

M034 · Impact / inverse measurement

# Ballistic Pendulum Lab

**Can you work out a projectile's speed from how high the pendulum swings?** Yes, if you work backward in two steps. The swing after the hit conserves energy, so the height the pendulum rises gives its speed just after impact. The hit itself loses energy but conserves momentum, and momentum turns that speed into the projectile speed. If the pendulum swings right over the top, its rise no longer tells you the speed.

A projectile embeds in a bob at the bottom of a massless rigid rod. The impact conserves horizontal momentum but loses mechanical energy. The nonlinear swing that follows conserves mechanical energy, even though the pivot pushes on the rod. Fit the first clear peak in the camera data to infer the projectile speed, then try a large swing or a full rotation.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The projectile embeds instantly and ideally at the bottom. The bodies are point masses on a massless rigid rod, and the swing has no friction. The rod keeps its constraint even through full rotations. The incoming flight is shortened on screen. The nonlinear motion is integrated with RK4, checked by an independent energy calculation and by a real solve at half the step size. The seven-frame local quadratic peak is an estimate limited by sampling; a rise too small to resolve, or a rotating bob, cannot give a speed from height. The camera records for ten seconds after contact.

Physics tutorial

## Two stages, two conservation laws

**Background** When the projectile embeds in the bob, the collision is strongly inelastic. Horizontal momentum sets the speed just after contact, and the swing that follows turns that kinetic energy into gravitational potential energy.

**Why it matters** OpenStax University Physics section 9.4 covers inelastic collisions, and section 15.4 covers pendulum motion. This apparatus combines the two without the small-angle approximation and measures the peak height from camera data.

Start with the essentials

- **Focus question**: Which conservation law applies during impact, and which applies while the rod swings?
- **One-sentence intuition**: The pivot pushes on the rod during the swing. Use horizontal momentum for the instant of impact at the bottom, then mechanical energy for the frictionless swing.

### Core mathematical model

#### Momentum at the instant of impact

$$
mu=(m+M)V,\qquad Q_{\mathrm{impact}}=\frac12\frac{mM}{m+M}u^2
$$

Embedding loses mechanical energy. The horizontal momentum relation applies at the moment of contact at the bottom.

#### Full pendulum equation for the rod

$$
\ddot\theta=-\frac gL\sin\theta,\qquad E=\frac12(m+M)L^2\dot\theta^2+(m+M)gL(1-\cos\theta)
$$

Keeping the exact sine term allows large swings. The rigid rod keeps the bob on its circle even through full rotations.

#### Speed from a measured turning height

$$
\widehat u=\frac{m+M}{m}\sqrt{2g\,\widehat h},\qquad\dot\theta(t_{\mathrm{peak}})=0
$$

The height comes from a local quadratic fit to seven frames around the peak, not from the energy prediction. Finite sampling and noise can bias it.

#### Height alone fails once the rod goes over the top

$$
V\ge2\sqrt{gL}\quad\Longrightarrow\quad h_{\max}=2L\ \text{does not uniquely identify }u
$$

At or above the separatrix, the swing has no finite first turning point that could identify the speed, so the instrument does not attempt the inversion.

### Common difficulties

#### Energy is lost in the impact

**Typical misconception** All the incoming kinetic energy turns into height.

**Better mental model** Embedding dissipates energy first. Only the kinetic energy left after impact drives the swing.

#### Momentum changes during the swing

**Typical misconception** The bob keeps its horizontal momentum throughout the arc.

**Better mental model** The pivot and gravity are external forces, so the momentum balance used at impact does not carry over to the swing.

#### A highest point is not always a turning point

**Typical misconception** Any top height gives a unique launch speed.

**Better mental model** A bob that goes all the way round passes the top with nonzero speed, so height alone is not enough.

### Run the experiment

1. 01

   #### Measure the first swing

   Choose Laboratory shot, press Record height trace and compare the fitted speed with the reference launch speed.

   **What to observe:** The local fit to the camera data can differ slightly from the reference because of noise and sampling.

2. 02

   #### Watch both stages

   Replay through the impact, then scrub through the swing.

   **What to observe:** The heat jumps at impact. During the swing, kinetic and potential energy trade back and forth.

3. 03

   #### Go beyond small angles

   Choose Large nonlinear swing and shorten the rod.

   **What to observe:** With the full sine equation the swing still conserves energy, until the motion reaches the threshold for full rotation.

4. 04

   #### Test when the speed can be found

   Try Full rotation, Rise below noise, a low frame rate and a smaller maximum integration step.

   **What to observe:** The instrument says when it cannot infer the speed. The energy error and the half-step solution check the numerical trajectory.

Physical conclusion

The speed estimate holds only if the projectile embeds, the setup is calibrated and the swing has a clear turning point. A rod that swings all the way round reaches the same top height for many different incoming speeds.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)