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.

Exact inspection time $0\,\mathrm{s}$

Downward weight magnitude $0\,\mathrm{N}$

Signed inward rod force $0\,\mathrm{N}$

Reference impact time $\text{Waiting}$

Amber ruler span $0\,\mathrm{m}$

Amber tick interval $0\,\mathrm{m}$

Acquired height frames $0$

Time from launch $0\,\mathrm{s}$

Reference incident speed $20\,\mathrm{m\,s^{-1}}$

Reference post-impact speed $0\,\mathrm{m\,s^{-1}}$

Incident speed inferred from camera peak $\text{Collecting}$

First resolved camera peak $\text{Collecting}$

Energy-predicted turning height $0\,\mathrm{m}$

Instantaneous rise $0\,\mathrm{m}$

Unwrapped rod angle $0{}^{\circ}$

Instantaneous mechanical energy $0\,\mathrm{J}$

Irreversible embedding loss $0\,\mathrm{J}$

Horizontal momentum defect during impact $0\,\mathrm{kg\,m\,s^{-1}}$

Mechanical plus heat balance defect $0\,\mathrm{J}$

Half-step endpoint phase displacement $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Local peak-fit residual RMS $\text{Collecting}$

Watch a collision, then infer what happened

Can you work out a projectile's speed from how high the pendulum swings?

Applied forcesBlue arrows show weight; violet arrows show the air or rod force. Force and velocity arrows each have their own scale, fixed for the whole trial. An instantaneous impact is not drawn as a finite force arrow.Metric rulerThe amber ruler uses the same length scale as the motion. Body sizes, frame thickness and the camera hardware are only schematic.Solved reference path so farThe green path is the solved physical reference; camera observations are recorded separately below.Velocity and contact vectorsArrows are schematic; readouts retain physical units.Projectile mass

m

0.08\,\mathrm{kg}

Bob mass

M

1.20\,\mathrm{kg}

Incident speed

u

20.00\,\mathrm{m\,s^{-1}}

Rigid rod length

L

1.00\,\mathrm{m}

Gravity

g

9.81\,\mathrm{m\,s^{-2}}

Maximum integration step

\Delta t_{\max}

4.00\,\mathrm{ms}

Camera frame rate

f_{\mathrm{cam}}

60.00\,\mathrm{Hz}

Height noise standard deviation

\sigma_h

0.00\,\mathrm{m}

Acquired trajectory

\tau

0.00\,\%

Drag the hollow bob at the release point to change the rod length. Record height trace takes new camera data. The fit uses only frames already captured, and it needs a rise and a curve that stand out clearly in the data. The height of the lowest point, the masses and gravity are calibrated separately. Once the bob swings all the way around, its height alone can no longer tell different incoming speeds apart.

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

BackgroundWhen 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 mattersOpenStax 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 misconceptionAll the incoming kinetic energy turns into height.

Better mental modelEmbedding dissipates energy first. Only the kinetic energy left after impact drives the swing.

Momentum changes during the swing

Typical misconceptionThe bob keeps its horizontal momentum throughout the arc.

Better mental modelThe 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 misconceptionAny top height gives a unique launch speed.

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

Run the experiment

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.
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.
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.
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.