---
title: "Newton’s Cradle & Contact Time Interactive Lab | Sandbox Physics"
description: "Does one ball in always mean exactly one ball out?"
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/newtons-cradle/"
last_modified: "2026-11-04"
---

M036 · Finite elastic contact

# Newton’s Cradle & Contact Time

**Does one ball in always mean exactly one ball out?** One ball in gives an almost isolated outgoing ball when equal masses touch elastically, but the rule is approximate. Several contacts compress at once, leaving small motions in the other balls. Unequal masses, gaps and dissipation can change the outgoing pattern substantially.

Slow down a Newton’s cradle collision. Change individual masses, gaps and stiffness to see overlapping contacts and departures from one-in, one-out.

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

Model assumptions

Mass is concentrated at each sphere centre; massless rigid rods constrain motion in a vertical plane and can push or pull. Intrinsic sphere spin, rod and frame collisions, air drag and plasticity are omitted. All sphere pairs interact through unilateral Hertz contact. Equal radii and independent effective masses describe idealized coated or core-loaded spheres; effective modulus includes both surfaces. Optional clipped Hunt–Crossley damping prevents tensile contact force; its coefficient has units of seconds per metre and is not a calibrated restitution coefficient. Contact overlap represents indentation, not interpenetrating rigid solids. The record ends when indentation reaches five percent of radius. Contact patches are see-through annotations of the Hertz area. Gravity and fixed supports exchange momentum; constraint forces do no work. Energy plus accumulated dissipation is checked. The replay repeats one computed first-impact and first-swing trial, with extra slowing during contact; it does not simulate long-term synchronization.

Physics tutorial

## A collision takes time

**Background** Pull back one ball and the far ball swings out. Slow down the impact: several contacts compress together before the last ball separates.

**Why it matters** The familiar rule works well for a nearly uniform elastic row. Finite compression explains its small departures and its larger failures when masses or gaps change.

Start with the essentials

- **Focus question**: Does one ball in always mean exactly one ball out?
- **One-sentence intuition**: The row carries a finite force pulse through several simultaneous contacts; separate two-body collisions are only an approximation.

### Core mathematical model

#### Elastic compression of two spheres

$$
F_{\mathrm H}=k\delta_+^{3/2},\qquad k=\frac43E^*\sqrt{R^*},\qquad R^*=\frac{R_1R_2}{R_1+R_2}
$$

Compression is the overlap of the undeformed spherical surfaces. The positive part gives zero force after separation. Effective modulus includes both surfaces.

#### A constrained bob feels every contact

$$
m_iL^2\ddot\theta_i=\mathbf F_i^{\mathrm{contact}}\cdot\frac{\partial\mathbf r_i}{\partial\theta_i}-m_igL\sin\theta_i
$$

Every pair contributes to the net contact force. Rod constraints keep the centres on circles; no instantaneous velocity-swap rule is applied.

#### Dissipation and the external supports

$$
F=F_{\mathrm H}\max(0,1+\alpha\dot\delta),\qquad E_{\mathrm{mech}}+Q=E_0,\qquad \Delta\mathbf P=\int\mathbf F_{\mathrm{external}}\,dt
$$

Compression speed is positive while the surfaces approach. Dissipated work accumulates in heat. Gravity and the fixed pivots exchange momentum even though ideal rod forces do no work.

### Common difficulties

#### Equal energy and momentum do not fix every velocity

**Typical misconception** Conservation alone forces exactly one ball to leave.

**Better mental model** A row has many outgoing velocities. The contact law and its finite duration select the motion; the default last ball receives most, but not exactly all, of the energy.

#### A replay is one finite trial

**Typical misconception** Each loop continues the previous oscillation.

**Better mental model** The same first-impact and first-swing record repeats. The contact stage has extra slow motion, and the clock reports physical time.

### Run the experiment

1. 01

   #### Resolve the pulse

   Start with One ball released. Pause during the gold contact glow and compare the rows below.

   **What to observe:** Several contact rows light up together; the first-pair force has a resolved half-height width.

2. 02

   #### Change the outgoing motion

   Try Two balls released, A heavy middle bob and Small gaps between balls. Click any bob to change its own mass.

   **What to observe:** The outgoing energy split and contact overlap change. Mass edits stay with the selected bob.

3. 03

   #### Separate stiffness from loss

   Increase Effective contact modulus, then try Dissipative contact.

   **What to observe:** Stiffer elastic contact gives a shorter pulse; damping also lowers the outgoing swing by converting mechanical energy to heat.

Physical conclusion

Read the pulse width and overlapping contact rows before treating the cradle as a sequence of isolated impacts.

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)