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Collision Mechanics Simulator

Perfectly Inelastic Collision Simulator — Momentum Conservation and Energy Loss

From the masses m_1, m_2 and velocities v_1, v_2 of two bodies, compute the common post-collision velocity v_f, the initial and final kinetic energies and the energy-loss fraction in real time. The body layout and a kinetic-energy bar chart make the "maximum-loss" collision tangible.

Parameters
Mass m_1
kg
Velocity v_1
m/s
Mass m_2
kg
Velocity v_2
m/s

Defaults are m_1 = 2.0 kg, v_1 = 10.0 m/s, m_2 = 3.0 kg, v_2 = 0.0 m/s — the canonical impact of a moving body onto a stationary one. They yield v_f = 4.0 m/s and a 60.0% energy-loss fraction. Set v_2 negative for a head-on collision, or same-sign for a rear-end one.

Results
—
Common final velocity v_f
—
Initial kinetic energy
—
Final kinetic energy
—
Energy-loss fraction
Bodies before and after collision

Top row = before (blue disk = m_1, orange disk = m_2, arrows = v_1, v_2) / bottom row = after (purple disk = merged m_1+m_2, arrow = v_f) / disk radii scale as mass^(1/3) (visual mass ratio) / arrow lengths scale with absolute speed

Kinetic-energy comparison

Blue bar = initial kinetic energy KE_i / green bar = final kinetic energy KE_f (after merging) / red bar = lost energy DeltaKE = KE_i - KE_f / numerical labels above each bar [J] / KE_f/KE_i complements the loss fraction printed in the top-right corner

Theory & Key Formulas

In a perfectly inelastic collision two bodies (masses $m_1, m_2$, velocities $v_1, v_2$) merge after impact and move together with a single common velocity $v_f$. Momentum is conserved exactly and kinetic energy is reduced by the largest possible amount.

Common velocity from momentum conservation:

$$v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$$

Initial and final kinetic energy:

$$KE_i = \tfrac{1}{2} m_1 v_1^{2} + \tfrac{1}{2} m_2 v_2^{2},\qquad KE_f = \tfrac{1}{2}(m_1+m_2)\,v_f^{2}$$

Energy loss expressed via the reduced mass $\mu = m_1 m_2/(m_1+m_2)$ and the relative motion:

$$\Delta KE = KE_i - KE_f = \tfrac{1}{2}\,\mu\,(v_1 - v_2)^{2}$$

$m_1, m_2$ are masses [kg], $v_1, v_2$ are signed velocities [m/s], and $v_f$ is the common post-collision velocity [m/s]. The limit $e = 0$ of the coefficient of restitution turns the missing kinetic energy into heat, sound, plastic deformation and adhesion.

What the Perfectly Inelastic Collision Simulator does

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A perfectly inelastic collision is the one where the two bodies stick together, right? What does that actually let me compute?
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Good entry point. It is the e = 0 limit of the coefficient of restitution: after impact the two bodies merge and move with a single velocity v_f. Momentum is rigorously conserved (whenever there are no external impulses), but kinetic energy is reduced by the largest amount any collision can lose. Try the defaults m_1 = 2 kg, v_1 = 10 m/s, m_2 = 3 kg, v_2 = 0 m/s. You should read v_f = (2 × 10 + 3 × 0)/5 = 4.0 m/s, initial KE = 100 J, final KE = 40 J and a 60% loss fraction.
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Momentum is conserved but energy is not? That sounds contradictory.
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A fair worry, but not a real contradiction. Momentum conservation only requires that the external impulse vanishes; internal forces (adhesion, plastic flow, friction) cancel in pairs and never shift the total. Kinetic energy, however, is a sum of squared speeds that is freely converted by internal forces into heat, sound and deformation. Algebraically you get DeltaKE = (1/2) m_1 m_2/(m_1+m_2) (v_1 - v_2)^2 — exactly the kinetic energy of the relative motion at the reduced mass mu. With the defaults mu = 2 × 3/5 = 1.2 kg and (10-0)^2 = 100, so DeltaKE = 60 J, matching the result panel.
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You said "maximum loss". How do I actually reach 100%? With the defaults I only see 60%.
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You hit 100% when the total momentum vanishes, i.e. m_1 v_1 + m_2 v_2 = 0. In this tool set m_1 = m_2 = 2 kg, v_1 = +10 m/s, v_2 = -10 m/s. You get v_f = 0, KE_i = 200 J, KE_f = 0 J, loss = 100% — two equal lumps of clay annihilate their motion and 100% of the kinetic energy becomes heat and deformation. In the centre-of-mass frame, the centre is already at rest, so when the merger happens there is no kinetic energy left to distribute. That is the deepest "intuition picture" for a perfectly inelastic head-on collision.
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Where does this matter in the real world? Just textbook problems?
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Plenty of real applications. The most famous is the ballistic pendulum: shoot a 0.01 kg bullet at 400 m/s into a 2 kg block, measure the post-impact velocity, and back out the muzzle speed. Set m_1 = 0.01, v_1 = 400, m_2 = 2, v_2 = 0 in this tool: v_f is about 1.99 m/s and the loss fraction is 99.5%. Automotive crumple zones are designed close to "perfectly inelastic" to absorb the occupant's kinetic energy through plastic deformation; NCAP-class frontal tests achieve a 96-99% loss fraction depending on the structure.
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In the bar chart KE_f is 40 J and DeltaKE is 60 J, summing to KE_i = 100 J. Is that a coincidence?
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No, it is the definition. Energy conservation in the full sense says KE_i = KE_f + DeltaKE_loss; the missing kinetic part has not vanished, it has become heat, sound, deformation and adhesion. Sweep v_1 in this tool and you will see KE_i and DeltaKE both grow as v_1^2, but the ratio (the loss fraction) depends only on the mass ratio and the relative velocity. Set v_1 = v_2 and the relative speed is zero, so the red bar collapses to zero and there is no loss at all.

Frequently Asked Questions

A perfectly inelastic collision is the limit case e = 0 of the coefficient of restitution, in which the two bodies merge and move with a single common velocity after impact. Momentum is conserved exactly, but kinetic energy is reduced by the largest possible amount, and the missing energy reappears as heat, sound, plastic deformation or adhesion. Unlike elastic (e = 1, energy-conserving) or partially inelastic collisions (0 < e < 1), the final velocity is determined uniquely by m_1, m_2, v_1 and v_2. Classic examples are colliding lumps of clay, vehicle crash zones, and a bullet embedding itself in a ballistic pendulum. Use this tool to vary the masses and velocities and see which combinations maximise the energy-loss fraction.
Conservation of momentum is rigorous whenever the impulse from external forces is zero; internal contact friction and adhesion cancel out and never change the system's total momentum. Kinetic energy, on the other hand, can be dissipated by those internal forces, because plastic deformation, acoustic radiation and frictional heat carry energy out of the mechanical degrees of freedom. Formally, two bodies with relative velocity (v_1 - v_2) lose exactly the kinetic energy (1/2) mu (v_1 - v_2)^2 of their relative motion, where mu = m_1 m_2/(m_1 + m_2) is the reduced mass. With the defaults m_1 = 2 kg, v_1 = 10 m/s, m_2 = 3 kg, v_2 = 0 m/s, this gives mu = 1.2 kg, relative-speed squared 100 m^2/s^2, DeltaKE = 60 J — exactly 60% of the initial KE_i = 100 J.
The fraction DeltaKE/KE_i grows with the share of kinetic energy that lives in the relative motion, which is maximised when the centre-of-mass velocity (and hence the total momentum) is small. A head-on collision with m_1 v_1 + m_2 v_2 = 0 gives v_f = 0 and a 100% loss — all kinetic energy goes into heat, sound and deformation. Conversely, two bodies moving with the same velocity have zero relative speed and zero loss. In this tool set v_1 = 10 m/s, v_2 = -10 m/s with equal masses to see the 100% case, and v_1 = v_2 to see the 0% case. Extreme mass ratios (e.g. a 0.01 kg bullet hitting a 10 kg block) also produce very large losses, near 99.9%.
The simulator assumes one-dimensional momentum conservation and a clean, frictionless merger; it ignores rotation, oblique impact, the finite contact time, body elasticity, fracture and temperature-dependent material properties. Real automotive crashes involve coupled plastic deformation of the frame, airbag work, occupant inertia and seatbelt forces, and require explicit finite-element solvers such as LS-DYNA, PAM-CRASH or Abaqus/Explicit. Even a ballistic pendulum needs corrections for air drag, rotational moment and rope tension. The tool is more than enough for first-principles understanding and order-of-magnitude estimates; for design-grade analysis switch to a dedicated commercial solver.

Real-World Applications

Ballistic pendulum and projectile-speed measurement: Benjamin Robins's 18th-century instrument hangs a block of mass M on a string and fires a bullet of mass m into it. Treating the bullet-block merger as perfectly inelastic gives V = mv/(m+M); the swing height h then yields V = sqrt(2 g h), from which the muzzle velocity v is recovered. With m = 0.01 kg, v = 400 m/s, M = 2 kg, v_2 = 0 this tool returns V about 1.99 m/s and a 99.5% loss. Modern labs use high-speed cameras and optical gates, but the inelastic-merger model still provides the reference calibration.

Automotive crumple zones: The front frame, bumper and subframe of a vehicle are designed to deform plastically and absorb the occupant's kinetic energy — behaviour close to perfectly inelastic. An NCAP 56 km/h frontal test treats the 1500 kg vehicle as colliding with an effectively infinite wall (m_2 -> infinity), giving essentially a 100% conversion of kinetic energy into vehicle deformation. Maintaining occupant survival space while maximising energy absorption is solved with explicit FEM (LS-DYNA) on a 10-ms time scale.

Cosmic dust and impact craters: Cosmic particles entering the atmosphere (10^-6 to 10^-3 kg at 11-72 km/s) decelerate and largely vaporise in essentially inelastic collisions with air molecules and the surface. Geophysics approximates the Chicxulub impactor (10 km diameter, 20 km/s) as a perfectly inelastic event, releasing of order 10^23 J as heat, seismic waves and ejecta. Set extreme mass ratios in this tool to watch most of the small body's kinetic energy be transferred to the large one.

Fixed-target particle accelerators: Fixed-target high-energy physics shoots a proton or electron at a stationary nucleus, and most of the beam energy is lost to "perfectly inelastic" final states whose centre-of-mass energy grows only as sqrt(2 m E). With extreme m_1 << m_2 (or m_1 >> m_2) in this tool, the energy fraction available to the centre-of-mass collapses. This is precisely the inefficiency that drove CERN to build the LHC as a collider rather than a fixed-target machine.

Common Misconceptions and Pitfalls

The first common misconception is that "a perfectly inelastic collision violates conservation of energy". It does not: the missing kinetic energy DeltaKE turns into heat, sound, plastic deformation and adhesion — non-mechanical forms of energy that the first law of thermodynamics still tracks faithfully. Classical mechanics says "mechanical energy is not conserved" because it only counts kinetic and potential, but the full energy is. In this tool DeltaKE is literally the energy that ends up as a dented bumper, a melting bullet or a frictional temperature rise, and it can be measured by thermometers and acoustic sensors in the lab.

The next misconception is that "a perfectly inelastic collision always brings the bodies to rest". That is only true when the total momentum is zero. In general v_f equals the centre-of-mass velocity (m_1 v_1 + m_2 v_2)/(m_1 + m_2), which can be substantial — with the defaults of this tool v_f = 4.0 m/s, an obvious motion. The correct mantra is "merger means motion at the centre-of-mass velocity", not "merger means stopping". Bodies come to rest after the impact only if they were already at rest in the centre-of-mass frame.

Finally, treating "a real vehicle collision as a single-point perfectly inelastic event" is dangerous. This tool is one-dimensional and assumes a point-mass merger, ignoring rotation, stiffness distribution, occupant restraints, fracture modes and the finite contact time. Real engineering uses explicit finite-element codes (LS-DYNA, Abaqus/Explicit) on meshes of 10^5 to 10^6 elements, with sheet-thickness, spot-weld and hardening-curve data. Treat the perfectly inelastic estimate as a starting point and always cross-check with a detailed FEM or with experimental crash data.

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