🚗 Crash Test — Impulse-Momentum Theorem

The vehicle buttons set a preset mass for a compact car, sedan or SUV; the mass slider then lets you fine-tune it from 800 to 3000 kg. The impact-speed slider sets the vehicle's speed on collision, from 10 to 130 km/h. The crumple-zone stiffness slider sets the effective spring stiffness of the front structure, from soft and energy-absorbing to stiff and nearly rigid, which the simulator converts into a stopping distance, stopping time and force profile.

The front structure is treated as a linear spring of stiffness k, so the deceleration follows simple harmonic motion: m·x″ = −k·x, where x is the crush depth. From energy conservation, ½mv₀² = ½k·x_max², the maximum crush is x_max = v₀√(m/k). The car comes to rest after a quarter period of that oscillation, Δt = (π/2)√(m/k), and the peak force at maximum crush is F_peak = k·x_max = v₀√(mk). This is a standard first-order model of a crumple zone: more realistic than assuming constant deceleration, because it produces a force that rises smoothly to a peak rather than jumping straight to an average value.

Because the crash force is not constant, it ramps up from zero to a maximum as the spring compresses, following a sine curve in time. For any simple harmonic crush, the average force over the stopping interval works out to exactly 2/π (about 64%) of the peak force. So the peak deceleration you feel at the moment of maximum crush is always noticeably higher than the average deceleration over the whole crash, which is why crash-safety limits are specified as peak g, not average g.

Tolerance depends heavily on direction, duration and restraint. Well-restrained occupants in modern cars can survive brief peaks of 40 to 60 g in a properly engineered structure, because seatbelts and airbags spread the force over the body and extend the occupant's own stopping time. Sustained deceleration above roughly 40 to 50 g for more than a few tens of milliseconds becomes dangerous to internal organs and the spine, which is exactly why vehicle structures, not just the occupant restraints, are engineered to keep structural peak g as low as possible.

The vehicle's crumple zone slows the car's structure, but the occupant inside is a separate mass that would otherwise keep moving at the original speed until something stops them abruptly. Seatbelts and airbags exist to apply the same impulse-momentum principle to the occupant: they extend the time and distance over which the person decelerates, so the same change in the occupant's momentum happens with far less peak force on the body than an unrestrained impact with the dashboard or windscreen would produce.

Real crash-test force-versus-crush curves are more complex than a single linear spring: they include nonlinear stiffening, structural buckling, and sometimes multiple stages as different parts of the car fail in sequence, and occupant kinematics inside the cabin are governed by separate restraint dynamics. This simulator uses the simple harmonic spring model because it is the simplest system that correctly shows a rising force with a genuine peak above the average, which is the core physics lesson. It is a first-order teaching model, not a substitute for real finite-element crash simulation.