---
title: "Two-Dimensional Collision Table Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/collision-table/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/collision-table/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## The geometry of an oblique collision

**Background** When two smooth disks touch, the impulse acts along the line joining their centers. Momentum is a vector, so the recoil must conserve momentum in both directions at once.

**Why it matters** OpenStax University Physics section 9.5 covers collisions in more than one dimension. On the air table you can see the normal impulse and compare velocities fitted from the camera with the exact contact solution.

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 misconception** With zero restitution, disks in a glancing hit must stick together.

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

#### A stopped disk has no direction

**Typical misconception** A disk that stops has a scattering angle of zero.

**Better mental model** In 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 misconception** Whenever the disks visibly touch, their velocities change.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

Physical conclusion

Adding up speeds cannot replace adding momentum vectors. The direction of the contact normal decides the impulse, the recoil and how much energy of relative motion is lost.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)