Set mass, velocity, and coefficient of restitution for two bodies to simulate 2D collisions. Explore momentum conservation via collision animation, velocity vector diagram, and energy budget chart across three tabs.
Parameter Settings
Body 1 Mass m₁
kg
Body 1 Speed v₁
m/s
Body 1 Angle θ₁
°
Body 2 Mass m₂
kg
Body 2 Speed v₂
m/s
Body 2 Angle θ₂
°
Coefficient of Restitution e
Results
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p₁ before (kg·m/s)
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p₂ before (kg·m/s)
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v₁' (m/s)
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v₂' (m/s)
0.0 J
ΔKE (Energy Loss)
Vec
Solid lines show pre-collision velocity; translucent dashed lines show post-collision velocity. Top: body 1 (blue), bottom: body 2 (red).
Anim
Two-sphere collision animation. Check how velocities change after impact.
Energy
Compare kinetic energy and momentum magnitudes before and after collision.
You say "momentum is conserved," but even if objects deform or generate heat at the moment of collision, is it really conserved?
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Momentum is conserved, but kinetic energy may not be. During a collision, the two bodies exert internal forces on each other. Those internal forces cancel as a pair, so total momentum does not change. But energy converted into deformation, heat, and sound is no longer kinetic energy, so KE can decrease (ΔKE < 0).
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I tried the "equal mass" preset with coefficient of restitution e=1, and the velocities of the two balls swapped completely. Is this always the case, not just a coincidence?
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Yes, for a head-on perfectly elastic collision with equal masses (e = 1), the equations give v₁' = u₂ and v₂' = u₁. That is why in billiards one ball can stop while the next leaves with nearly the same speed. Newton's cradle shows the same idea.
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When I set the "perfectly inelastic collision" preset (e=0), the two objects stuck together and moved as one. ΔKE showed a large negative value. How much energy is lost?
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For equal masses in a head-on perfectly inelastic collision, the bodies stick together and move with the center-of-mass velocity. If their speeds are equal and opposite, that velocity is zero, so all initial kinetic energy is dissipated into deformation, heat, and sound. In vehicle crashes, crumple zones are designed to absorb that energy intentionally.
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When I selected the "oblique collision" preset, the velocity vectors after collision became angled. How is this calculated?
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An oblique collision is solved by decomposing motion into the collision normal and tangential directions. Momentum conservation and the restitution equation apply along the normal direction; without friction, the tangential velocity component does not change. This simulator uses the x direction as a simplified collision normal.
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Does the motion after collision differ between hitting a heavy object with a light one, versus hitting a light object with a heavy one?
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The mass ratio matters a lot. When a heavy body hits a much lighter one, the heavy body keeps nearly the same speed while the light body can leave much faster. When a light body hits a heavy one, it mostly rebounds. Try the m₁ = 8, m₂ = 1 preset to see the effect.
Frequently Asked Questions
Are there cases where momentum conservation does not hold?
It does not hold when external forces (gravity, friction, etc.) are not negligible. However, if the collision time is extremely short (impact force >> external force), it approximately holds. Also, for charged particles in an electromagnetic field, the field itself carries momentum, so particle momentum alone is not conserved (total system momentum is conserved).
How can the coefficient of restitution be measured?
The simplest method is a "drop test." Drop a ball from height h₀ and measure the rebound height h₁; then e = √(h₁/h₀). A superball has e ≈ 0.9, a tennis ball e ≈ 0.7, and a steel ball on a steel plate e ≈ 0.5. Temperature also affects it; a cold ball has a lower e.
Can explosions (fragmentation) also be solved using momentum conservation?
Yes. An explosion is the reverse of a collision. If a stationary object (total momentum = 0) explodes into two fragments, from m₁v₁ + m₂v₂ = 0 we get v₂ = -(m₁/m₂)v₁, so the two fragments fly in opposite directions. Rocket propulsion (ejecting gas backward to obtain reaction) is also an application of momentum conservation.
Why do objects become slowest when they stick together after collision?
In a perfectly inelastic collision (e = 0), the relative velocity after collision is zero (they stick together and move at the same speed). From momentum conservation, v' = (m₁u₁ + m₂u₂)/(m₁ + m₂). This equals the center-of-mass velocity (total momentum divided by total mass), and this collision loses the maximum energy. The energy loss is ΔKE = -μ(u₁ - u₂)²/2, where μ is the reduced mass.
What software is used for real-world traffic accident simulation?
LS-DYNA (Ansys) and Radioss (Altair) are industry standards. They use explicit finite element methods to compute complex deformation, fracture, airbag deployment, and dummy behavior during crashes. Computation time: a real-time collision of 10–100 ms takes several hours to simulate. Automakers validate against actual crash tests (NCAP) before using the results in design.
Can a superball have a coefficient of restitution greater than 1?
For a pure coefficient of restitution, e ≤ 1 is a physical constraint (e > 1 would mean kinetic energy increases after collision, violating energy conservation). However, mechanical systems (e.g., balls with spring mechanisms, explosive bolts) can exhibit apparent e > 1 behavior. The maximum e for a superball is about 0.92, never reaching 1.
What is 2D Collision & Momentum Conservation?
22D Momentum & Collision Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations of 2. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind 2 are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.