Drag a ball to pull it up, release it, and watch the elastic collision chain propagate. Tweak gravity, string length, and restitution coefficient to feel how energy flows.
Parameters
Gravity g
m/s²
String Length L
m
Restitution e
Speed Multiplier
×
Presets
Display Options
Drag the leftmost ball upward, then release to start.
Results
0.00
Kinetic Energy J
0.00
Potential Energy J
0.00
Total Energy J
—
Period T (s)
Cradle
Energy Time History
CAE Connection
The foundational model for impact simulations in LS-DYNA and Abaqus/Explicit. Modeling the restitution coefficient is essential in impact energy absorption design and crash safety analysis. Ball mills and stamping processes in manufacturing are governed by the same contact mechanics.
What exactly is Newton's Cradle demonstrating? I see the balls swing and click, but what's the core physics principle?
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Basically, it's a brilliant visual demo of conservation of momentum and elastic collisions. When you lift and release one ball, its momentum transfers through the line and launches the ball on the opposite end. In this simulator, you can adjust the Restitution (e) slider to see what happens when the collision isn't perfectly elastic.
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Wait, really? So if I set the Restitution to less than 1, the balls won't swing as high? What's the math behind that?
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Exactly! A perfect elastic collision (e=1) conserves both momentum and kinetic energy. But in real life, some energy is lost as heat or sound. The restitution coefficient 'e' quantifies that loss. For instance, try setting 'e' to 0.7. You'll see the outgoing ball speed is only 70% of the incoming speed, so it won't swing as high.
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That makes sense. What about the other controls, like Gravity and String Length? They seem to change the swinging speed.
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Great observation! The Gravity (g) and String Length (L) control the pendulum motion before the collision. A higher gravity pulls the ball down faster, shortening the swing period. A longer string makes the swing slower and more graceful. Play with these sliders to see how they affect the timing of the impacts, which is crucial for synchronizing collisions in engineering applications.
Physical Model & Key Equations
The motion of each ball before collision is governed by the pendulum equation. For small angles, it approximates simple harmonic motion.
$$\ddot{\theta}= -\dfrac{g}{L}\sin\theta$$
Where $\theta$ is the angular displacement, $g$ is gravitational acceleration, and $L$ is the string length. The simulator uses Verlet integration to solve this equation numerically for stable, realistic swinging.
The core of the cradle is the collision model. For two equal-mass balls, the post-collision velocities ($v_1'$, $v_2'$) are determined by conservation of momentum and the restitution coefficient.
$$
\begin{align*}m v_1 + m v_2 &= m v_1' + m v_2' \quad \text{(Momentum Conservation)}\\[4pt]
e &= -\frac{v_1' - v_2'}{v_1 - v_2}\quad \text{(Definition of Restitution)}\end{align*}$$
Solving these gives the simple rule: $v_1' = \frac{(1-e)v_1 + (1+e)v_2}{2}$ and $v_2' = \frac{(1+e)v_1 + (1-e)v_2}{2}$. For $e=1$ (perfectly elastic), the balls simply exchange velocities. For $e\lt 1$, some kinetic energy is lost.
Frequently Asked Questions
Drag the ball up to a sufficient height and release the mouse. If the lift angle is too small, the collision chain will be weak. Also, if the coefficient of restitution is less than 1, energy will dissipate and the chain is more likely to break. Try setting the coefficient of restitution to 1.0.
The coefficient of restitution (e) determines the velocity ratio after collision. e=1 means a perfectly elastic collision, and e=0 means a perfectly inelastic collision (objects stick together). In CAE, it is used for designing impact absorbers (reducing e to absorb energy) and evaluating the performance of crash safety components.
Increasing gravity (g) increases the falling speed of the balls, resulting in higher collision energy. Increasing the string length (L) lengthens the pendulum period, widening the interval between collisions. These correspond to parameters in CAE for adjusting collision timing and impact force.
This tool simplifies elastic collisions of equal masses for display, whereas LS-DYNA and Abaqus/Explicit analyze contact between complex shapes and materials using the finite element method. The concept of the coefficient of restitution and the law of conservation of momentum are common, allowing intuitive understanding of the fundamental concepts of impact analysis.
Real-World Applications
Crash Safety Simulation (CAE): This is the foundational model for impact simulations in software like LS-DYNA and Abaqus/Explicit. Engineers model the restitution coefficient of car bumpers, airbags, and crumple zones to design vehicles that absorb crash energy effectively and protect passengers.
Ball Mill Operations: In mining and material processing, large rotating drums filled with metal balls crush ore. The efficiency of this grinding process depends on the elastic and inelastic collisions between the balls, directly governed by the same momentum principles.
Stamping and Forging Processes: In manufacturing, when a hammer strikes a metal workpiece, the deformation and energy transfer are analyzed using collision models with a restitution coefficient. This helps in designing machinery that shapes metal efficiently without wasting energy.
Sports Equipment Design: The "trampoline effect" in golf clubs, tennis rackets, and baseball bats is essentially a collision problem. Engineers optimize materials to achieve a desired restitution (e.g., a high "coefficient of restitution" or COR in golf) to maximize ball speed and distance.
Common Misconceptions and Points to Note
First, are you assuming that "since there are 5 balls, pulling one from the end will make only one ball move from the opposite side"? Actually, this is only true for the ideal case of perfectly symmetric initial conditions and perfectly elastic collisions (e=1). For example, if you initially pull and release two balls simultaneously, two balls will fly out from the opposite side. This is necessarily determined by the conservation of momentum and energy. Try it in the simulator. Next, a pitfall in parameter settings. If you set "Gravity" to 0, the balls will move in a straight line at constant speed after a collision and won't return. This is because they cease to behave as pendulums. This is a classic example of how setting extreme parameter values in practical work can cause the model to break down. Finally, the misconception that "the coefficient of restitution e is determined solely by the material." In reality, it also varies with impact velocity, temperature, and surface condition. When performing collision analysis in CAE, it's common to use velocity-dependent models for e rather than treating it as a constant. The key is to observe the behavioral changes when you modify e in this tool, connecting them to the "percentage of energy loss" rather than just simple "damping."
Set gravitational acceleration (default 9.81 m/s²) using gravNum input
Enter cradle length in meters via lenNCNum (typical range 0.2–1.5 m)
Specify coefficient of restitution (0–1) in restNum; use 0.95 for steel balls, 0.85 for composite
Click simulate to observe momentum transfer between spheres and track Kinetic Energy, Potential Energy, Total Energy (J), and Period T (s) in real time
Worked Example
Steel Newton's cradle with 5 balls, each 0.5 kg: Set gravNum=9.81 m/s², lenNCNum=0.5 m, restNum=0.95. Release left ball from 45° displacement. Initial PE≈2.45 J converts to KE≈2.33 J at bottom (95% restitution). After elastic collision, right ball exits with nearly identical velocity. Period T≈1.42 s. Energy loss per swing ≈0.12 J due to friction and air resistance.
Practical Notes
Real Newton's cradles lose 2–5% energy per cycle; set restitution <0.98 for authentic damping behavior
Lengthen cradle strings for slower oscillation and clearer momentum observation in educational settings
Use restNum=1.0 only for idealized frictionless analysis; industrial demonstrations require 0.90–0.96
Monitor Total Energy decay to validate gravitational potential conversion and collision efficiency