M033 · Scattering / vector momentum

Two-Dimensional Collision Table

Aim a disk past a stationary target and drag its impact parameter. Follow the contact normal, recoil and outgoing tracks; the momentum polygon closes in two dimensions. Camera fits recover normal restitution. A stopped disk, a tangent touch and a clean miss expose different measurement limits.

Elapsed time $0\,\mathrm{s}$

Horizontal momentum in camera frame $0\,\mathrm{kg\,m\,s^{-1}}$

Kinetic energy in camera frame $0\,\mathrm{J}$

Dissipated impact energy $0\,\mathrm{J}$

Delivered normal impulse $0\,\mathrm{N\,s}$

Predicted contact time $\text{No event}$

Restitution from camera velocities $\text{Collecting}$

Camera position residual RMS $\text{Collecting}$

Momentum conservation defect $0\,\mathrm{kg\,m\,s^{-1}}$

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

Vertical momentum $0\,\mathrm{kg\,m\,s^{-1}}$

Projectile velocity angle $0{}^{\circ}$

Target recoil angle $\text{Stopped}$

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

RELEASE / OBSERVE / INFER

What controls the scattering angle?

Velocity and contact vectorsArrows are schematic; readouts retain physical units.First mass

m_1

1.00\,\mathrm{kg}

Second mass

m_2

1.00\,\mathrm{kg}

Incident speed

u

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

Signed impact parameter

b

0.45\,\mathrm{m}

Normal restitution

e

1.00\,

Camera frame rate

f_{\mathrm{cam}}

60.00\,\mathrm{Hz}

Position noise standard deviation

\sigma_{\mathrm{pos}}

0.00\,\mathrm{m}

Acquired trajectory

\tau

0.00\,\%

Drag the blue launch marker up or down. Blue and amber tracks share a spatial scale; velocity components are plotted separately. Angles are signed from the positive horizontal axis; a stopped disk has no direction. Camera estimates are not clipped into the physical interval.

Smooth frictionless disks on an unbounded air table, fixed physical radii and no spin. The impulse follows the line of centers and preserves relative tangential velocity. Zero normal restitution does not glue disks together in a glancing impact. Tangent contact transfers no impulse. The apparatus camera follows the disks with a metric ruler; the track plot uses equal spatial scales. Camera timing, scale and contact normal are independently calibrated.

Start with the essentials

Focus question: Why can zero restitution still leave two disks moving apart after a glancing collision?
One-sentence intuition: Restitution constrains the relative normal component; the relative tangent is unchanged because the contact is frictionless.

Core mathematical model

Closing normal and impulse

g_n=(\mathbf u_1-\mathbf u_2)\cdot\mathbf n<0,\qquad J=-\frac{(1+e)g_n}{1/m_1+1/m_2}

The normal points from the target center toward the incident disk at contact. A separating or tangent pair receives no impulse.

Two-dimensional velocity update

\mathbf v_1=\mathbf u_1+\frac{J}{m_1}\mathbf n,\qquad\mathbf v_2=\mathbf u_2-\frac{J}{m_2}\mathbf n

Equal and opposite impulses conserve both momentum components.

Normal restitution and tangent

(\mathbf v_1-\mathbf v_2)\cdot\mathbf n=-e\,g_n,\quad(\mathbf v_1-\mathbf v_2)\cdot\mathbf t=(\mathbf u_1-\mathbf u_2)\cdot\mathbf t

Zero restitution removes only the normal relative motion; it does not impose adhesion.

Geometric contact and energy loss

|\mathbf r_1(t_c)-\mathbf r_2(t_c)|=R_1+R_2,\qquad\Delta K=\frac12\frac{m_1m_2}{m_1+m_2}(1-e^2)g_n^2

Exact linear motion determines the first contact root. A beam beyond the sum of radii misses; a tangent has no normal impulse.

Common difficulties

Zero restitution is not glue

Typical misconceptionZero restitution must merge glancing disks.

Better mental modelKeep a nonzero offset: unchanged relative tangent makes the pair separate.

A stopped vector has no angle

Typical misconceptionA stopped projectile has a zero scattering angle.

Better mental modelEqual-mass head-on elastic exchange stops it; the direction readout reports stopped.

Touch is not impulse

Typical misconceptionEvery visible touch changes velocity.

Better mental modelExact tangency has no closing normal speed and no momentum transfer.

Run the experiment

01
Scatter and close the polygon
Record the glancing elastic preset.
What to observe: The two outgoing momentum arrows add to the original incident arrow.
02
Change the geometry
Drag the blue launch marker through positive and negative offsets.
What to observe: The contact normal and signed deflection reverse with the offset.
03
Dissipate only the normal
Choose zero normal restitution at nonzero offset.
What to observe: Tangential relative motion remains even though the normal opening speed vanishes.
04
Find the boundary
Compare head-on, tangent and miss presets.
What to observe: A stopped disk, a zero-impulse touch and a miss have distinct geometric meanings.