stock here, I was never sure whether the relative deceleration ( and thus human damage, would be twice as much or 4 times as much)
lets talk specifics of a 6000 lb vehicle hitting 3000 lb vehicle head on. there is conservation of kinetic energy, and conservation of momentum. Also include 2 more scenarios, in which the collision is elastic or inelastic, and make a rational basis for the elastic using modern post 2010 vehicles in USA. What is the deceleration of vehicle A compared to vehicle B (the 3000 pounder)
ChatGPT said:
Great question. I’ll label the 6,000-lb vehicle A (≈2,720 kg) and the 3,000-lb vehicle B (≈1,360 kg). Speeds will be taken as equal and opposite to make things concrete (e.g., both at vv toward each other). You can swap in different speeds if you like.
Core physics (1-D head-on)
- Momentum is conserved in every case.
- Kinetic energy is only conserved in a perfectly elastic collision. Real crashes are largely inelastic (energy goes into crush, heat, sound, etc.).
- During contact the force on both cars is equal and opposite, so their instantaneous accelerations are inversely proportional to mass:
∣aA∣:∣aB∣=mB:mA∣aA∣:∣aB∣=mB:mA.
With 2:1 mass ratio, B sees ~2× the acceleration/deceleration of A for the same crash pulse.
1) Perfectly inelastic (they lock together)
Final speed (to the right positive):vf=mAv+mB(−v)mA+mB=mA−mBmA+mB v=13vvf=mA+mBmAv+mB(−v)=mA+mBmA−mBv=31v
Change in velocity (ΔvΔv, which sets deceleration severity):∣ΔvA∣=∣vf−v∣=23v,∣ΔvB∣=∣vf−(−v)∣=43v∣ΔvA∣=∣vf−v∣=32v,∣ΔvB∣=∣vf−(−v)∣=34v
So B’s ΔvΔv is 2× A’s (and so is its average/peak decel, for similar pulse duration).
2) Perfectly elastic
Final speeds:vA′=mA−mBmA+mBv+2mBmA+mB(−v)=−13v,vB′=2mAmA+mBv+mB−mAmA+mB(−v)=53vvA′=mA+mBmA−mBv+mA+mB2mB(−v)=−31v,vB′=mA+mB2mAv+mA+mBmB−mA(−v)=35v∣ΔvA∣=∣−13v−v∣=43v,∣ΔvB∣=∣53v−(−v)∣=83v∣ΔvA∣=−31v−v=34v,∣ΔvB∣=35v−(−v)=38v
Again B’s ΔvΔv is 2× A’s.
When is “elastic” a rational model?
- Very low-speed bumper taps (parking speeds) where modern post-2010 cars use energy absorbers/bumper beams that spring back with minimal permanent deformation approximate an elastic collision.
- At moderate/high road speeds, modern crumple zones intentionally make the crash highly inelastic to lower peak g’s—so use the inelastic model as the realistic bound.
Worked numbers (so you can gauge g-loads)
Take both cars at 30 mph each (closing speed 60 mph). v=13.41 m/sv=13.41 m/s.
- Inelastic:
∣ΔvA∣=8.94 m/s∣ΔvA∣=8.94 m/s, ∣ΔvB∣=17.88 m/s∣ΔvB∣=17.88 m/s. - Elastic:
∣ΔvA∣=17.88 m/s∣ΔvA∣=17.88 m/s, ∣ΔvB∣=35.76 m/s∣ΔvB∣=35.76 m/s.
If the crash pulse lasts ~0.12 s (typical order of magnitude), average decelerations are:
- Inelastic:
aA≈75 m/s2≈7.6 gaA≈75 m/s2≈7.6g,
aB≈149 m/s2≈15.2 gaB≈149 m/s2≈15.2g. - Elastic (not realistic at this speed, but a bound):
aA≈15.2 g, aB≈30.4 gaA≈15.2g,aB≈30.4g.
Key takeaway: with equal and opposite approach speeds, the lighter 3,000-lb car experiences about twice the ΔvΔv and about twice the deceleration of the 6,000-lb car, regardless of whether the collision is modeled as elastic or perfectly inelastic. Real post-2010 vehicles are engineered so the event is predominantly inelastic (energy absorbed by crush) to reduce the peak g’s for occupants.
Great—let’s pin it down with your numbers.
I’ll call the 6,000-lb car A (~2,720 kg) and the 3,000-lb car B (~1,360 kg). Both at 55 mph (24.587 m/s) toward each other.
What “% inelastic” means
Crash “elasticity” is captured by the coefficient of restitution eee:
- e=0e=0e=0 = perfectly inelastic (lock together).
- e=1e=1e=1 = perfectly elastic (full rebound).
Real 55 mph car crashes are highly inelastic: e≈0e\approx 0e≈0–0.1.
“25% inelastic” would mean e=0.75e=0.75e=0.75 (i.e., 75% elastic) — that’s not realistic at road speeds. I’ll show three cases so you can see the sensitivity.
Δv (sets severity) and average decel (depends on pulse time)
For both cars at 55 mph, the change in velocity magnitudes ∣Δv∣|\Delta v|∣Δv∣ are:
| Assumption | ∣ΔvA∣|\Delta v_A|∣ΔvA∣ (m/s) | ∣ΔvB∣|\Delta v_B|∣ΔvB∣ (m/s) |
|—|—:|—:|
| e=0e=0e=0 (lock, inelastic) | 16.391 | 32.783 |
| e=0.10e=0.10e=0.10 (slightly elastic, realistic) | 18.031 | 36.061 |
| e=0.75e=0.75e=0.75 (“25% inelastic”, unrealistic) | 28.685 | 57.370 |
Rule that always holds: B’s Δv\Delta vΔv ≈ 2× A’s (because force is equal and masses differ 2:1).
To convert Δv\Delta vΔv to average deceleration, divide by the crash pulse duration:aˉ=ΔvΔt,gˉ=aˉg\bar a = \frac{\Delta v}{\Delta t},\qquad \bar g=\frac{\bar a}{g}aˉ=ΔtΔv,gˉ=gaˉ
Modern belt+airbag “ride-down” stretches the occupant pulse to ~0.15–0.20 s; the vehicle structure pulse is often ~0.10–0.12 s. Here are two useful views:
Using a 0.12 s vehicle pulse (structure)
| Assumption | A avg g | B avg g |
|---|---|---|
| e=0e=0e=0 | 13.93 g | 27.86 g |
| e=0.10e=0.10e=0.10 | 15.32 g | 30.64 g |
| e=0.75e=0.75e=0.75 | 24.34 g | 48.68 g |
Using a 0.18 s occupant pulse (belts/airbag ride-down)
| Assumption | A avg g | B avg g |
|---|---|---|
| e=0e=0 | 9.29 g | 18.57 g |
| e=0.10e=0.10 | 10.21 g | 20.43 g |
| e=0.75e=0.75 | 16.25 g | 32.50 g |
Bottom line
- At 55–55 head-on, a realistic assumption is highly inelastic (e≈0e≈0 to 0.1), not “25% inelastic.”
- Under realistic e≈0.1e≈0.1 and typical occupant pulse ~0.18 s, the average decel to the passenger is roughly:
- A (6,000 lb): ~10 g
- B (3,000 lb): ~20 g
- B experiences about twice the deceleration of A for the same pulse, due to the 2:1 mass ratio.
If you give me a pulse duration you prefer (e.g., data from a specific crash or sled test), I’ll plug it in and return exact g’s for A and B.
ChatGPT said:
Yes — if we’re saying 10 g or 20 g average deceleration, the timeframe is already baked in because that “g” figure came from ΔvΔv divided by the crash pulse duration.
That means we’re talking about an average load over the time the occupant is decelerating (e.g., 0.15–0.20 s for a belted occupant in a modern car). In reality, the peak g’s can be 1.5–2× higher than the average because the decel curve is not perfectly flat — it ramps up and down.
Human tolerance context
A healthy adult in good shape can tolerate quite different levels depending on direction, restraint, and duration:
| Magnitude | Duration | Likely Effects |
|---|---|---|
| Up to ~5 g | 0.2 s | Very survivable; discomfort, possible belt bruising. |
| 10 g (avg) | 0.15–0.20 s | Belt marks, minor contusions, potential mild whiplash; low fracture risk in healthy adults. |
| 20 g (avg) | 0.15–0.20 s | High belt force: cracked ribs in some occupants, moderate whiplash, shoulder bruising; increased concussion risk if head strike. |
| 35–45 g (peak) | 0.05–0.10 s | Rib and clavicle fractures common; internal organ strain possible; high concussion probability even with airbags. |
| >50 g (peak) | <0.05 s | Severe thoracic injuries, organ rupture, major fractures; often fatal without exceptional restraint systems. |
Joints and Musculoskeletal Effects
- 10 g avg over ~0.18 s: Ligaments and tendons are unlikely to fail if forces are spread through the restraint system. Most damage comes from secondary impacts (knees to dash, wrists/hands braced on steering wheel).
- 20 g avg over ~0.18 s:
- Seat belt loads on clavicle/sternum can cause rib or collarbone fractures in older adults.
- Knee impacts can fracture patella/tibia if unrestrained.
- Neck sees higher shear force → greater whiplash and cervical strain risk.
Internal Organ Effects
The body’s soft tissues have inertia too:
- Lungs: At ~20 g avg, belt compression and chest decel can bruise lung tissue (pulmonary contusion), especially in older or smaller-framed occupants.
- Heart & Aorta: Rapid decel causes shear between heart/aorta and surrounding tissues. At peaks above ~35–40 g, risk of aortic rupture sharply rises.
- Brain: Even if skull doesn’t strike anything, acceleration can cause concussion. Risk rises above ~15–20 g average when peaks go higher.
Bones
- 10 g avg: Low fracture probability except for brittle bones (osteoporosis) or if secondary impacts occur.
- 20 g avg: Moderate fracture risk from restraint forces (esp. ribs, sternum, clavicle), plus extremity fractures from intrusion or flailing.
Summary for your crash case:
- Vehicle A (10 g avg) → Likely survivable for healthy occupants; minor to moderate restraint injuries.
- Vehicle B (20 g avg) → Survivable in many cases with belts/airbags, but much higher risk of rib/clavicle fractures, organ bruising, and concussion.
now run it for 75MPH, say in the UP, but a glancing blow, head on, but the bumpers overlap by 10% so there will be spinning afterwards and lateral accelerations also. This get tricky I know it, so think long and big
ChatGPT said:
Here’s a first-order run at your 75 mph each glancing, small-overlap (≈10%) head-on.
I’ll call the 6,000-lb vehicle A (≈2,720 kg, ~2.0 m wide) and the 3,000-lb vehicle B (≈1,360 kg, ~1.8 m wide).
Modeling assumptions (kept simple but physics-based)
- Small-overlap ~10% → line of action passes near an outer corner (≈0.9 m from A’s CG, ≈0.81 m from B’s CG), so the impact produces strong yaw (spin) as well as longitudinal decel.
- High-speed crash is highly inelastic: coefficient of restitution e≈0.05e≈0.05.
- No tangential (friction) impulse at the contact (conservative for longitudinal Δv; lateral loads then come mainly from yaw).
- Typical yaw inertias IzIz from vehicle size; occupant pulse ≈0.18 s (belt+airbag ride-down).
- All values are order-of-magnitude, not a reconstruction.
Results
Pre-impact speeds: +75 mph (A), –75 mph (B)
Longitudinal (forward/back) outcomes
- Δv (change in speed)
- A: ~17.6 m/s (39.5 mph)
- B: ~35.3 m/s (78.9 mph)
- Average occupant decel (0.18 s pulse)
- A: ~10 g (peaks commonly 15–20 g)
- B: ~20 g (peaks commonly 30–40 g)
- Post-impact forward speed
- A: ~35.6 mph forward
- B: ~3.9 mph forward
The 2:1 mass ratio still shows up: B takes ~2× the longitudinal Δv and g.
Yaw / spin and lateral loading (from the small-overlap)
- Yaw rate jump right after contact
- A: ~390 °/s
- B: ~805 °/s
- (If nothing else stops them, the first 0.5 s would spin about 195° (A) and ≈400° (B)—in reality tire forces and secondary hits will clamp these.)
- Average lateral g from the initial spin
(estimated ≈ vpost⋅Δω/2gvpost⋅Δω/2g)- A: ~5–6 g (peaks ≈ 8–12 g)
- B: ~~1–2 g (peaks ≈ 2–3 g; B’s forward speed is almost killed, so lateral from yaw is smaller unless there’s tangential bite or a secondary hit)
- Resultant average g (vector of long.+lat.)
- A: ~11–12 g
- B: ~~20 g
What this says physically
- The light car (B) still sees about double the frontal loading of A.
- Small-overlap at 75–75 produces violent yaw, especially for B. That shifts some loading into oblique/lateral directions—bad for head/neck and for side-structure contact.
- Real-world injury risk will be dominated by peaks and secondary contacts (guardrail/trees/another car). At these speeds, intrusion becomes likely in the small-overlap side, especially for the lighter car, which can overwhelm the restraint “ride-down” benefit.
cause of this impact direction, would they be more likely or less likely to slip out of the 3 point restraint system, and what would the results then be?
ChatGPT said:
Short answer: with a 10% small-overlap, glancing head-on, occupants are not likely to fully slip out of a modern 3-point belt, but the loading direction does increase the odds of partial “belt escape”—either the shoulder belt sliding off the shoulder or the lap belt riding up (submarining)—especially in the lighter car (B) that sees ~2× the longitudinal g and in the car that spins harder.
Why oblique/small-overlap raises the risk
- Yaw + lateral acceleration rotate the torso across the belt (far-side motion). The belt is optimized for straight-ahead frontal loading; oblique paths let the shoulder belt slip toward the neck or off the shoulder.
- Seat rotation and pelvis ramping during small-overlap can let the lap belt slide above the iliac crests → submarining.
- Load limiters and retractor spool-out (good for chest injury in straight frontal) can increase forward/lateral excursion in an oblique hit.
What happens if partial escape occurs
- Shoulder belt off the shoulder (far-side):
- Head swings wide → higher chance of A-pillar, dash, or far-side interior contact despite airbags.
- More neck shear/bending; higher concussion risk.
- Torso may still be held by the lap belt → fewer abdominal injuries than submarining, but more head/neck issues.
- Lap belt ride-up / submarining:
- Abdominal organ and bowel injuries (“seat-belt syndrome”), possible lumbar spine flexion fractures.
- If the shoulder belt stays engaged, chest loads continue; if it also slips, head contact risk spikes.
- In extreme cases with intrusion + rotation, partial out-of-position airbag strike can worsen facial/neck injury.
Why complete ejection is still unlikely
- Post-2010 cars usually have: dual pretensioners (retractor + buckle), anti-submarining seat pans/ramps, knee airbags, side torso + curtain airbags, sometimes center/far-side airbags.
- Doors and glazing plus 3-point restraint make full belt escape/ejection rare unless there’s gross misuse (belt under the arm/behind the back), severe seatback recline, or major intrusion that physically defeats the belt path.
Which car is at greater risk here?
- Vehicle B (3,000 lb): higher longitudinal Δv and g → more forward excursion → more chance the lap belt rides up if posture is poor.
- The vehicle that spins harder (often B in your scenario) also sees more far-side shoulder-belt slip because the torso is flung laterally.
Practical ways to reduce escape risk (everyday use)
- Sit upright (seatback ~100–110°), hips all the way back; do not recline.
- Lap belt low and tight on the pelvic bones; remove bulky coats; avoid belt-routing “under the arm.”
- Adjust D-ring so the shoulder belt crosses mid-shoulder (not near the neck, not off the shoulder).
- Keep head restraint close and level with the top of the head.
- For kids/short adults, use the proper booster/seat so the lap belt stays on the pelvis.
- Ensure airbags/pretensioners aren’t disabled after a prior event; fix any airbag light.
If someone does slip the belt
Expect patterns:
- Off-shoulder: facial lacerations, concussion, cervical sprain/strain; possible contact with pillar/dash/steering rim despite airbags.
- Submarining: abdominal bruising/guarding, intestinal/mesenteric injury, lumbar spine compression/flexion fractures; sometimes hip/pelvic injuries from belt edge.