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Category Car Calculators

Vehicle 1

Mass (kg)

Speed (km/h)

Direction angle (°)

Vehicle 2

Mass (kg)

Speed (km/h)

Direction angle (°)

Collision Settings

Collision Type

Perfectly Inelastice = 0, vehicles stick Partially Inelastic0 < e < 1 Elastice = 1, KE conserved

Collision Duration (ms) Typical: 80 – 150 ms

Presets:

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About

Vehicle crash reconstruction relies on conservation of linear momentum and the coefficient of restitution e to determine post-impact velocities and forces. Errors in mass estimation or pre-impact speed produce squared errors in kinetic energy calculations, which directly affect insurance liability assessments and forensic conclusions. This calculator solves the one-dimensional and angled two-body collision problem for elastic, perfectly inelastic, and partially inelastic cases. It computes ΔV (the change in velocity each vehicle experiences), peak impact force via the impulse-momentum theorem F = mΔv ÷ Δt, and total energy dissipated as deformation and heat. Results assume rigid-body approximation and do not model crumple zone nonlinearities or occupant kinematics.

Formulas

The post-collision velocities for a one-dimensional two-body collision with coefficient of restitution e are derived from simultaneous conservation of momentum and the restitution definition.

Conservation of momentum:

m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′

Coefficient of restitution:

e = − v₂′ − v₁′v₂ − v₁

Solved final velocity for vehicle 1:

v₁′ = m₁v₁ + m₂v₂ + m₂e(v₂ − v₁)m₁ + m₂

Impact force via impulse-momentum theorem:

F = m ⋅ ΔvΔt

Energy dissipated:

ΔKE = 12m₁v₁² + 12m₂v₂² − 12m₁v₁′² − 12m₂v₂′²

Where m₁, m₂ are vehicle masses in kg; v₁, v₂ are pre-impact velocities in m/s; v₁′, v₂′ are post-impact velocities; e is the coefficient of restitution (0 = perfectly inelastic, 1 = perfectly elastic); Δt is the collision duration in s; Δv is the velocity change (Delta-V) for each vehicle.

Reference Data

Vehicle Type	Typical Mass kg	Crash Rating Source	Crumple Zone Depth m	Notes
Subcompact Car	900 - 1100	Euro NCAP / NHTSA	0.3 - 0.5	Low mass increases ΔV
Compact Sedan	1200 - 1400	Euro NCAP / IIHS	0.4 - 0.6	Most common collision participant
Mid-Size Sedan	1400 - 1700	NHTSA 5-Star	0.5 - 0.7	Moderate energy absorption
Full-Size Sedan	1700 - 2000	IIHS Top Safety Pick	0.5 - 0.8	Higher momentum at same speed
Compact SUV	1500 - 1800	Euro NCAP	0.5 - 0.7	Higher center of gravity affects rollover
Full-Size SUV	2200 - 2800	NHTSA	0.6 - 0.9	Significant mass advantage
Pickup Truck	2000 - 3000	IIHS	0.5 - 0.8	Frame construction differs from unibody
Minivan	1800 - 2200	Euro NCAP	0.5 - 0.7	Passenger protection priority
Sports Car	1200 - 1600	Euro NCAP	0.3 - 0.5	Low profile complicates underride
Electric Vehicle (mid)	1800 - 2300	NHTSA / Euro NCAP	0.5 - 0.7	Battery pack adds floor mass
Electric SUV	2400 - 3100	IIHS	0.6 - 0.8	Heaviest consumer segment
Motorcycle	150 - 350	N/A	0.0 - 0.1	No crumple zone; rider ejection risk
Commercial Van	2500 - 3500	Euro NCAP Van	0.5 - 0.7	Payload varies total mass significantly
Light Truck (Box)	3500 - 7500	FMVSS	0.3 - 0.5	Rigid frame, minimal deformation
Semi-Truck (Loaded)	15000 - 36000	FMVSS 223/224	0.2 - 0.4	Underride guards mandatory; extreme momentum

Frequently Asked Questions

Delta-V (ΔV) is the total change in velocity a vehicle undergoes during impact. NHTSA research shows strong correlation between ΔV and occupant injury severity (AIS scale). A frontal ΔV above 25 km/h typically produces moderate injuries (AIS 2+) in belted occupants. Above 60 km/h, fatality probability exceeds 50%. This calculator computes ΔV for both vehicles from the momentum equations and restitution coefficient.

The coefficient of restitution e quantifies how much kinetic energy survives the collision. For real vehicle crashes, e typically ranges from 0.05 to 0.3 depending on structural stiffness, impact speed, and overlap geometry. At highway speeds (> 80 km/h), e approaches 0 because crumple zones fully compress. Using e = 1 (elastic) is physically unrealistic for cars and overestimates post-impact velocities while underestimating energy dissipation.

Conservation of momentum dictates that the impulse (F ⋅ Δt) is equal and opposite for both vehicles. Since ΔV = impulse ÷ mass, the lighter vehicle gets a proportionally larger velocity change. A 1000 kg car hitting a 3000 kg truck receives 3× the ΔV. This is why mass mismatch crashes (car vs. SUV) produce disproportionate injury to the lighter vehicle's occupants.

The force calculated via F = mΔv ÷ Δt represents the average force during the collision pulse. Real crashes produce a force-time curve with a peak roughly 1.5 to 2.5 times the average. Collision duration Δt for car-to-car impacts typically ranges from 80 to 150 ms. For car-to-rigid-barrier, 50 to 100 ms. The calculator uses the duration you specify, so selecting a realistic value is critical for force accuracy.

Yes. When approach angles differ from 0° (head-on) or 180° (same direction), the calculator decomposes velocities into x and y components using cos and sin. A 90° intersection collision transfers momentum perpendicular to one vehicle's travel direction, which produces lateral ΔV - the most dangerous for occupants because side structures offer less crumple zone depth (typically 0.15 - 0.25 m vs. 0.5 - 0.8 m frontal).

Collision duration depends on vehicle stiffness and impact speed. Typical values: car-to-car frontal 100 - 150 ms; car-to-barrier 60 - 100 ms; car-to-truck 120 - 200 ms; low-speed bumper contact 150 - 300 ms. Shorter durations produce higher peak forces. If unknown, 120 ms is a reasonable default for moderate-speed frontal impacts.