M034 · Impact / inverse measurement

Ballistic Pendulum Lab

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 conserves mechanical energy while the pivot exerts external force. Fit the first resolved camera peak to infer the projectile speed, then test a large swing or a complete rotation.

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 $\text{Collecting}$

RELEASE / OBSERVE / INFER

Can a height measurement identify launch speed?

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 release bob to change rod length. Record height trace takes fresh camera data; fitting uses only acquired frames and requires resolved curvature and rise. The bottom-height datum, masses and gravity are independently calibrated. Once the bob rotates, height alone cannot distinguish incoming speeds.

Ideal instantaneous embedding at the bottom; point masses, a massless rigid rod and no swing friction. The rod remains constrained through rotations. Incoming flight is visually compressed. Nonlinear RK4 dynamics have an independent energy audit and a true half-step solve. A seven-frame local quadratic peak is a finite-sampling estimate; unresolved rise and rotating states cannot yield speed from height. The camera records ten seconds after contact.

Start with the essentials

Focus question: Which conservation law applies during impact, and which applies while the rod swings?
One-sentence intuition: The pivot supplies external force during the swing. Use horizontal momentum for the instantaneous bottom impact, then mechanical energy for the frictionless swing.

Core mathematical model

Instantaneous embedding

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

Mechanical energy is lost during embedding. The horizontal momentum relation applies at the bottom contact.

Nonlinear rod dynamics

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

The exact sine force permits large swings. A rigid rod maintains the constraint through rotation.

Invert a resolved turning height

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

The height is a seven-frame local quadratic peak, not the energy-predicted height. Finite sampling and noise can bias it.

Rotation makes height insufficient

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

At the separatrix or above it, there is no finite first turning point that identifies speed. The instrument refuses the inversion.

Common difficulties

Do not conserve impact energy

Typical misconceptionAll incident kinetic energy becomes gravitational height.

Better mental modelEmbedding dissipates energy first; only the remaining post-impact kinetic energy feeds 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; the momentum balance used at impact cannot be extended across the swing.

A highest point is not always a turning point

Typical misconceptionAny top height gives a unique launch speed.

Better mental modelA rotating bob crosses the top with nonzero speed, so height alone is insufficient.

Run the experiment

01
Measure the first swing
Record the laboratory shot and compare the fitted speed with the launch reference.
What to observe: The local camera fit can differ slightly from the reference because of noise and sampling.
02
Inspect the two balances
Replay through impact, then scrub the swing.
What to observe: Heat jumps at impact while kinetic and potential energy exchange during the swing.
03
Leave the small-angle limit
Choose large nonlinear swing and shorten the rod.
What to observe: The sine dynamics still conserve swing energy until the rotation boundary is reached.
04
Test identifiability and integration
Try rotation, rise below noise, low frame rate and a smaller maximum step.
What to observe: The instrument reports failure honestly; the energy defect and half-step solve audit the numerical trajectory.