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.

Infer projectile speed from a recorded pendulum rise. Separate momentum at impact from energy during the swing, and expose the error of conserving energy through embedding.

Speed inferred from camera $20\,\mathrm{m\,s^{-1}}$

Measured turning height $0.0795\,\mathrm{m}$

Reference speed after impact $1.25\,\mathrm{m\,s^{-1}}$

Energy lost at impact $15\,\mathrm{J}$

Model assumptions

The projectile travels in an ideal horizontal guide and embeds instantly at the lowest point. Two parallel, massless rigid guides keep the catcher horizontal, so its kinetic energy is translational. Gravity and ideal constraints act during the frictionless nonlinear swing. Guides may sustain compression; rotations are ideal mathematical continuations without apparatus contact. Shapes and catcher mass distribution are schematic. Incoming flight is replayed eightfold slower; swing time is unchanged. The camera records three seconds with independent Gaussian height noise. A seven-frame quadratic peak fit uses acquired samples; unresolved peaks, missing turning points and rotations cannot identify speed. RK4 is checked against energy and a separately integrated half-step solution.

Focus question: Why does one measured height lead to two very different speed estimates?
One-sentence intuition: Work backward: the frictionless swing gives the speed immediately after impact. Momentum at the bottom then gives the incoming speed.

Core mathematical model

Momentum during embedding

mu=(M+m)V,\qquad Q=\frac{mM}{2(M+m)}u^2

At the bottom the ideal guide impulse is vertical, so horizontal momentum is conserved during the instantaneous impact. Embedding dissipates mechanical energy.

Energy during the swing

\frac12(M+m)V^2=(M+m)gh,\qquad \ddot\theta=-\frac{g}{L}\sin\theta

This first equality applies between the instant after impact and a true turning point. The full sine equation permits large swings; the catcher stays horizontal.

Correct inference and the energy-only mistake

\widehat u=\frac{M+m}{m}\sqrt{2g\widehat h},\qquad u_{\mathrm{wrong}}=\sqrt{\frac{2(M+m)g\widehat h}{m}}

Both estimates use the same fitted camera height. The second wrongly assigns all incoming kinetic energy to the rise. For the default masses it gives about five metres per second instead of twenty.

When height cannot identify speed

V\geq 2\sqrt{gL}\quad\Longrightarrow\quad \sup_{t\ge0}h(t)=2L

At or above the separatrix, there is no finite first turning point. Different rotating trajectories share the same top height. A noise-limited or unrecorded peak is also withheld from the inference.

Common difficulties

The pivot is an external support

Typical misconceptionThe swing conserves horizontal momentum too.

Better mental modelThe guide force has a horizontal component away from the bottom. It changes momentum while doing no work in the ideal constraint.

A recorded peak has uncertainty

Typical misconceptionThe displayed reference curve is the measured data.

Better mental modelDots are simulated noisy camera samples. The dashed curve comes from the ideal trajectory. A seven-frame local fit estimates the first resolved peak from the dots.

Run the experiment

01
Compare the two answers
Start with Small swing, large loss. Follow the projectile and compare the speed estimates below.
What to observe: Most energy disappears from mechanical motion at contact; the energy-only answer is much too small.
02
Design the rise
Pause and drag the catcher upward, then resume. Try Comparable masses.
What to observe: The incident speed changes with the designed height. A heavier projectile retains a greater fraction of its initial kinetic energy.
03
Find the measurement limits
Try Over the top and Rise buried in noise. Open the settings to change camera noise or frame rate, then take a new trial.
What to observe: An unresolved or non-invertible peak removes the speed estimate. Each new trial changes the camera samples while retaining the ideal reference.