Real-time animation of elastic and inelastic collisions in 1D and 2D. Adjust coefficient of restitution to experience momentum conservation and energy loss. Visualize impulse with force-time graphs.
Mode
Mass m₁ (kg)1.0
Mass m₂ (kg)1.0
Velocity v₁ (m/s)3.0
Velocity v₂ (m/s)-1.0
Restitution e1.0
Presets
Pre/Post Collision Values
4.00
p_total (kg·m/s)
5.50
KE total (J)
0.00
KE loss (J)
0.00
Impulse J (N·s)
Impulse-Momentum Theorem
$$J = \Delta p = m\Delta v = F \cdot \Delta t$$
Restitution: $e = \frac{v_2'-v_1'}{v_1-v_2}$
CAE Note: Automotive crash analysis (LS-DYNA) extends collision duration through crumple zones to reduce peak force on occupants while keeping impulse constant.
Impact Force F(t) vs Time (Impulse = area under curve)
What is Impulse & Collision?
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So, in this simulator, I can make two balls collide. What exactly is the "coefficient of restitution" slider controlling?
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That's the key! Basically, the restitution 'e' tells you how "bouncy" the collision is. It's the ratio of the relative speed *after* the collision to the relative speed *before*. In practice, if you set e=1, it's a perfectly elastic collision—like two super bouncy balls. Try sliding it to 0 and see what happens; the balls will stick together perfectly.
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Wait, really? So if momentum is always conserved, what's actually being lost when e is less than 1?
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Great observation. Momentum ($m\vec{v}$) is *always* conserved in the collision if we consider the whole system. But kinetic energy ($\frac{1}{2}mv^2$) is only fully conserved when e=1. For e<1, that "lost" kinetic energy is converted into other forms. For instance, in a car crash, it becomes heat, sound, and the energy used to crumple the metal. Try setting a very low 'e' with high velocities in the simulator—you'll see a big slowdown after impact.
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That makes sense. So the "Impulse" mentioned is the change in momentum, right? How is that connected to the force during the crash?
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Exactly right. Impulse ($J$) is the change in momentum, and it's also equal to the *average* collision force multiplied by the collision time: $J = F_{avg}\Delta t$. This is the crucial link for safety engineering. If you keep the impulse constant, you can reduce the peak force on passengers by *increasing* the collision time $\Delta t$. This is why cars have crumple zones! In the simulator, the impulse is calculated for you—watch how it changes as you adjust masses and velocities.
Physical Model & Key Equations
The fundamental principle governing all collisions is the Conservation of Linear Momentum. For a system with no external forces, the total momentum before the collision equals the total momentum after.
Here, $m_1, m_2$ are the masses, $\vec{v}_1, \vec{v}_2$ are the initial velocities, and $\vec{v}_1', \vec{v}_2'$ are the final velocities after the collision. This equation always holds true for the collision itself.
The Impulse-Momentum Theorem connects the force during the collision to the change in motion of an object. The impulse applied to an object causes its momentum to change.
$$\vec{J}= \Delta \vec{p}= m \Delta \vec{v}= \vec{F}_{avg}\cdot \Delta t$$
$\vec{J}$ is the impulse vector, $\Delta \vec{p}$ is the change in momentum, and $\vec{F}_{avg}$ is the average force over the collision duration $\Delta t$. This shows that for a given change in momentum, a longer collision time results in a smaller, safer average force.
The nature of the collision is defined by the Coefficient of Restitution (e): $e = -\frac{v_2' - v_1'}{v_2 - v_1}$. It ranges from 0 (perfectly inelastic, stick together) to 1 (perfectly elastic, kinetic energy conserved).
Real-World Applications
Automotive Crash Safety & Crumple Zones: This is a direct application of the impulse-momentum theorem. Cars are designed with front and rear sections that crumple in a controlled way. This increases the collision time (Δt) during a crash, which dramatically reduces the average force (F_avg) on the passenger compartment for the same impulse, saving lives. CAE software like LS-DYNA is used to simulate and optimize these crumple zones.
Sports Equipment Design: The padding in helmets (for football, cycling) or the strings in a tennis racket are designed to increase the time over which an impact occurs. When a ball hits a racket or a head hits the ground, the longer Δt means a lower peak force is transmitted to the player's hand or skull, preventing injury.
Ballistics & Impact Testing: Engineers use these principles to design armor and test materials. By analyzing the momentum change and restitution of a projectile hitting a plate, they can calculate the average impact force and determine if the material will fail. This is critical for military, aerospace, and protective gear.
Particle Physics & Astrophysics: At both the smallest and largest scales, collision laws apply. In particle accelerators, scientists analyze collisions to discover new particles, conserving momentum in their calculations. In astrophysics, the equations govern everything from the formation of planets (accretion) to the colossal collisions of galaxies.
Common Misconceptions and Points to Note
There are a few key points you should be especially mindful of when starting to use this simulator. First is that the coefficient of restitution e is not determined solely by material. It's common to memorize things like "steel has e=0.9, clay has e=0," but in reality, it changes significantly with impact velocity and object shape. For example, even the same steel ball will see a decrease in 'e' due to localized plastic deformation if it collides at ultra-high speeds. While the simulator lets you set it as a single parameter, keep in mind that real-world CAE uses much more complex models.
The second point is not to confuse "conservation of momentum" with "balance of forces". At the instant of collision, the two objects exert equal and opposite forces on each other according to Newton's third law. However, this equality refers to the "force," not the "change in momentum," which differs if the masses are different. When a heavy ball (m1=10kg) collides with a light ball (m2=1kg), the lighter one changes its velocity much more dramatically, right? The change in momentum (impulse) is equal for both, but the change in velocity is inversely proportional to the mass.
Finally, a common pitfall in the simulator settings is the realism of initial conditions. For instance, setting an extremely large mass (1000kg) and a high velocity (100m/s) might work mathematically, but in reality, it would generate tremendous energy, shattering the objects. Since the goal here is to understand the principles, the trick to correctly grasping the phenomena is to experiment within a realistic range, like ball masses of 1–10 kg and velocities of a few meters per second.
Related Engineering Fields
The principles of this 1D/2D collision form the foundation for a surprisingly wide range of engineering fields. First on the list is automotive crash safety (crashworthiness). The relationship between impulse and peak force you see in the simulator is at the very core of vehicle body design. By designing the front end to crumple in stages, the collision time is extended from tens to hundreds of milliseconds, mitigating impact on occupants. CAE tools like LS-DYNA and RADIOSS simulate these multi-dimensional collisions in detail.
Next, rocket stage separation is also an application of collision and momentum. When separating the first and second stages of a rocket, small explosive bolts or springs push them apart. Since these are internal forces, momentum is conserved: the lighter upper stage accelerates significantly, while the heavier lower stage moves away slowly. You can easily visualize this phenomenon in the simulator by setting an extreme mass ratio for the collision.
Another field not to be overlooked is powder technology (particle technology). To analyze processes like mixing drug particles in pharmaceutical plants or collisions of fine particles in cement manufacturing, a simulation method called the Discrete Element Method (DEM) is used. It calculates the collision (coefficient of restitution, friction) of each of countless particles to predict macroscopic powder flow and mixing degree. Our simulator lets you experience the very basic physics of a "single collision," which is the foundation of this.
For Further Learning
Once you're comfortable with the basics using this tool, try taking the next step. First, I recommend tackling 2D collisions (oblique collisions). Switch the tool to 2D mode and try colliding balls at an angle. At that moment, the law of conservation of momentum holds independently for the x and y components. The coefficient of restitution formula is also applied only to the velocity component perpendicular to the collision surface. For example, when a ball hits a smooth floor at an angle, the horizontal velocity component remains unchanged, and only the vertical component changes according to the coefficient of restitution. Understanding this allows you to explain the motion of billiard balls from first principles.
Mathematically, finding the velocities before and after collision involves solving a system of equations from the momentum conservation law and the coefficient of restitution formula. It's simple in 1D, but requires vector calculations in 2D. A powerful concept here is thinking in terms of the "relative velocity vector after rebound". Consider the normal vector of the collision surface and process only the component along it with the coefficient of restitution. Textbooks on planar motion of rigid bodies are helpful for learning this.
If you want to go even further, try proving the conservation of kinetic energy in perfectly elastic collisions. When the coefficient of restitution e=1, you can derive that the sum of kinetic energy before and after the collision is conserved, starting from the momentum conservation law $$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$$ and the coefficient of restitution formula $$v_2' - v_1' = - (v_2 - v_1)$$. Working through this calculation yourself once will give you a deeper understanding of the relationship between energy and momentum. Once you can do that, the next topics expand into areas like "3D rigid body collisions considering angular momentum" and "CAE of collisions involving plastic deformation."