M033 · Scattering / vector momentum

Two-Dimensional Collision Table

What sets the scattering angle in a two-dimensional collision? The direction of the contact normal, set by how far off-center the hit is, and the restitution coefficient. The impulse acts only along the normal: restitution sets the relative velocity along it after impact, and without friction the relative velocity along the contact surface stays the same.

Aim a disk at a stationary target on an air table, drag the impact parameter, and turn the three-dimensional view to watch from any side. Follow the contact normal, the recoil and the outgoing tracks, and see the outgoing momentum vectors add up to the incoming one. Fitting the camera tracks recovers the normal restitution. A stopped disk, a grazing touch and a clean miss each limit the measurement in a different way.

Center of mass, horizontal $0\,\mathrm{m}$

Center of mass, transverse $0\,\mathrm{m}$

Center-of-mass horizontal velocity $0\,\mathrm{m\,s^{-1}}$

Relative speed on calibrated normal $\text{No event}$

Relative speed on calibrated tangent $\text{No event}$

Camera frames through the cursor $1$

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 $0\,\mathrm{J}$

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

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

Target recoil angle $\text{Stopped}$

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

Watch a collision, then infer what happened

What sets the scattering angle in a two-dimensional collision?

Past trajectoriesDraws the analytic reference path up to the cursor time.Center-of-mass markerMarks the mass-weighted average position in the chosen observer's frame.Delivered contact eventOutlines from the moment of contact and the impulse arrows appear only after the contact has happened.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. The blue and amber tracks use the same distance scale, and the velocity components are plotted separately. Angles are measured with a sign from the positive horizontal axis; a stopped disk has no direction. Camera estimates are shown as measured, even when they fall outside the physical range.

The disks are smooth and frictionless, with fixed radii and no spin, on an air table with no edges. The impulse acts along the line of centers, and the relative velocity along the tangent does not change. In a glancing impact, zero normal restitution does not make the disks stick together. An exactly tangent touch transfers no impulse. The camera view follows the disks and shows a metric ruler, and the track plot uses the same scale on both axes. The camera timing, the length scale and the contact normal are calibrated separately.

Start with the essentials

Focus question: Why can zero restitution still leave two disks moving apart after a glancing collision?
One-sentence intuition: Restitution sets only the relative velocity along the normal; the relative velocity along the tangent stays the same because the contact is frictionless.

Core mathematical model

Approach speed along the normal, and the 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}

At contact, the normal points from the target's center toward the incoming disk. Disks that are moving apart or only just touching receive no impulse.

Velocities after impact in two dimensions

\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.

Restitution along the normal, no change along the 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 relative motion along the normal; it does not make the disks stick.

When contact happens and how much energy is lost

|\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

The disks move in straight lines, so the first root of the contact equation gives the contact time. An offset larger than the sum of the radii misses; an exactly tangent touch has no normal impulse.

Common difficulties

Zero restitution does not glue the disks

Typical misconceptionWith zero restitution, disks in a glancing hit must stick together.

Better mental modelWith a nonzero offset, the relative velocity along the tangent does not change, so the disks still separate.

A stopped disk has no direction

Typical misconceptionA disk that stops has a scattering angle of zero.

Better mental modelIn a head-on elastic hit between equal masses, the incoming disk stops. A zero velocity has no direction, so the Lab reports the disk as stopped.

A touch does not always give an impulse

Typical misconceptionWhenever the disks visibly touch, their velocities change.

Better mental modelAt an exactly tangent touch, the disks have no approach speed along the normal, so no momentum is transferred.

Run the experiment

01
Find the instant of impact
Step through Just before impact, At impact and Just after impact, rotate the view, then switch to Conservation ledger.
What to observe: Positions change smoothly, but the velocities jump. The violet arrows show impulses already delivered at impact, not forces acting throughout the playback.
02
Follow the center of mass
Set unequal masses and watch the teal center-of-mass marker as you drag through the record.
What to observe: Its velocity stays constant even when the disks' velocities change. The momentum and energy totals cover only the time up to the cursor.
03
Check that momentum adds up
Choose Glancing elastic impact and press Record collision.
What to observe: The two outgoing momentum arrows add to the original incident arrow.
04
Change the offset
Drag the blue launch marker through positive and negative offsets.
What to observe: When the offset changes sign, the contact normal flips and the deflection reverses.
05
Remove only the normal motion
Choose Zero normal restitution with a nonzero offset.
What to observe: The disks no longer separate along the normal, but their relative motion along the tangent remains.
06
Compare the edge cases
Compare the Head-on exchange, Exact tangent and Miss the target presets.
What to observe: A disk that stops, a touch with zero impulse and a miss mean different things geometrically.

Core mathematical model

Approach speed along the normal, and the impulse

Velocities after impact in two dimensions

Restitution along the normal, no change along the tangent

When contact happens and how much energy is lost

Common difficulties

Zero restitution does not glue the disks

A stopped disk has no direction

A touch does not always give an impulse

Run the experiment

Find the instant of impact

Follow the center of mass

Check that momentum adds up

Change the offset

Remove only the normal motion

Compare the edge cases