I started out considering total energy dissipated only (hence my sqrt(2) above), but I don’t think that’s quite right. If (a) a car going 120 mph hits an identical stationary car, and the two stick together, and keep going at 60 mph (b) until they skid to a stop, then I think the energy in (a) mostly goes in to deformation and such, but (b) goes mostly into the brakes, tires, guardrail scraping, etc. So I think both matter, but that the two should be considered separately.
I don’t think its proper to assume they’ve stopped in tje 120 vs 0 scenario. 60 vs 60 they stop, just like hitting a wall. 120 vs 0, they both continue at 60mph in the direction the 120 Car was moving.
Crumple zones, friction of the road stoppong the cars, etc are all separate issues that distract from the model proposed in the OP. Assuming a circular cow, a frictionless road, a wall of infinite mass, etc… two cars hitting each other 60 is equivalent to one hittimg a wall at 60, one traveling at 120 hittimg a car of equal mass going 0, one going 110 hitting one going 10, they’re all equivalent with respect to mass and velocity
…on a treadmill…
The question is pretty much high school physics. The thing to work with is not just the kinetic energy in the system, you need to consider the momentum. Indeed the momentum is much easier to deal with, as unlike the kinetic energy, it isn’t subject to conversion to other forms (such as heat.)
Momentum is conserved. Before and after the crash the momentum in the system is the same.
Two cars of equal mass, speed and opposite direction have an aggregate momentum of zero. After they hit the momentum will sum to zero still. It doesn’t matter if the collision is elastic or not. Car at 120mph and a stationary car, both cars of same mass. After collision momentum is conserved - if they stick together is a jumbled wreck, they will be travelling at 60mph.
Car travelling at 60mph and the planet at rest. After the collision the total momentum is the same. The planet will have gained an imperceptible speed in the direction the car was travelling.
A car accelerating or slowing down is a system with the Earth, and as momentum is conserved, as the car accelerates or decelerates, so does the earth. The energy involved mostly just continues its long journey into low grade heat.
Change in momentum isn’t what kills you directly. Nor is it simply the energy. Acceleration does, but it is complex as the time you are accelerated for matters. That does mean that the energy expended matters, but not as a simple metric. However as a simple rule of thumb - the energy has to go somewhere, and apart from bending lots of metal the deceleration you experience does work on your body components. Enough acceleration for long enough is going to perform work such as disassembling vital plumbing in your chest or tearing fissures across your brain. Of course if enough metal bending occurs you may simply get caught up in the general changes in the geometry of the vehicle and suffer fatal changes to your own geometry. Part of the point of crumple zones and air bags and the like is to extend the time the deceleration occurs for enough that the peak loads are lower then the threshold for damage. The same amount of energy is dissipated into you, but over a time your body can cope with (we hope.) Crumple zones also isolate the geometric changes to somewhere safely away from you.
The amount of kinetic energy doesn’t matter. You have a bunch of kinetic energy when you’re just cruising down the highway, and it doesn’t hurt you at all. We’d be closer if we said that the amount of energy dissipated matters, but that’s still not quite right: When you get off the highway, you put on the brakes and slow, perhaps eventually to a stop, dissipating your kinetic energy, and that’s harmless, too. What really matters is the amount of energy that’s dissipated quickly.
When a car going 120 hits a stationary car, there’s more kinetic energy initially than when you have two cars going 60 hitting head-on, but not all of it is dissipated quickly. The two cars will both be moving at 60 after the crash, which means they still have a lot of their kinetic energy. That kinetic energy also will be eventually dissipated, but it’ll be dissipated gradually, like when you slam on your brakes, not quickly, like in the crash. The amount of energy dissipated in the crash itself will be the same for the 120-0 crash as for the 60 head-on.
The initial question was 2 cars at 60 vs 1 car against a wall at 120. Those are not the same. Assuming equal mass cars and inelastic collisions.
The car vs the wall, assuming a non deforming wall, is absorbing four times the energy in that scenario.
That is correct, if you’re talking about a wall.
That’s correct, and I don’t think anyone in this thread questioned that. The only question is how to treat any kinetic energy that remains after the initial collision. That was zero in both cases of the OP’s question, but then I complicated things by mentioning the case of vehicles with different mass.
That energy does eventually get dissipated, since all crashes end, one way or another, with both vehicles stationary. It’s clearly bad, since I’d rather be in a crashed car that’s stopped than in an identical crashed car that’s still skidding down the highway at 60 mph, but it’s probably not as bad as the energy lost in the collision, since you have some chance of dissipating that energy slowly and safely in the tires/brakes, vs. quickly in the crumple zones or your skull.
I think the question of how you weight those two different ways of losing energy is no longer simple physics. Ignoring “energy lost while skidding to a stop” probably understates the seriousness of a crash, but considering the two ways with equal weight probably overstates it.
How about this, as a way to inform the OP’s students about the relative consequences of collisions…
#1 Hitting a solid, immovable wall is twice as bad as hitting a parked car.
#2 A head-on collision with another car going the same speed as you is also twice as bad as hitting a parked car.
This assumes the both cars have equal mass.
It is also worth noting that in scenario #2 you have HALF as much time to recognize the impending collision and take action to avoid it.
What does it mean to be “twice as bad”?
Rapidly releases twice as much energy. Energy released rapidly isn’t a perfect measure of damage, but it’s a pretty good proxy (much better than, say, momentum).
As noted in earlier threads, the Mythbusters test really showed what happened. (Jamie had mistakenly mentioned off-hand the wrong result and viewers called him on it.)
The real shame was that the didn’t show both crashes at the same time the right way. Two cars on the top, one car vs. wall on the bottom. If you cover up the appropriate side of the screen it would awfully darn clear they were effectively the same. The impact point isn’t moving is both cases.
My God!
Former physics teacher here…
This is basic high school physics. If two vehicles (of equivalent mass) are each traveling at 60 mph and hit head on, it is essentially equivalent to one of the vehicles hitting an immovable wall at 60 miles an hour. That is because in both cases, the vehicles come to an immediate stop, and in both cases, the change in momentum (for any one of the cars) is the same.
This is NOT the same scenario as a vehicle traveling at 60 mph hitting a parked vehicle, because in this case, the crumpled mass of the two cars continue to move after the collision.
Here’s another extensive discussion:
The Mythbusters story in #2 is badly written, and while it’s basically correct, it seems to be trying to say that both versions are somewhat true but not clearly explaining why.
To those who claim that a head-on collision with both cars going 60 mph is the same as hitting a brick wall at 60, consider the case of car going 60 mph and hitting a brick wall that moving towards it at 60 mph. Clearly the relative speed is going to be 120 mph, exactly the same as if the wall was standing still and the car was going 120 mph.
The difference with vehicle collisions is that while the kinetic energy dissipated is the same as one car going 120 mph – the speeds are indeed additive – the cars’ energy-absorbing crumple zones help to dissipate the energy and the impact damage to each car. There is no basis for claiming that the damage will necessarily be the same as a 60 mph impact with a brick wall, and it almost certainly won’t be. It will just be a lot less than hitting a brick wall at 120 mph, and much depends on the design and mass of the vehicles involved. If there are two vehicles going 60 mph in opposite directions and one of them is a fully loaded cement truck and the other is a small car, the small car will experience something a lot like the 120 mph brick wall collision.
It all depends on whether you are using real world variation … (eg car headon with truck) or making rather unlikely “lab conditions” , like “two identical vehicles”.
Car head on with truck, its like car at high speed into a wall, truck damaged like it went 20 mph into a wall. Well of course the truck is heavier and the energy from it goes into damaging the car, so the damage can be like it was 100mph into the wall, or 300 mph into the wall …
two identical cars, hitting directly head on , its like each car 60 mph into wall separately.
variations allow situations in between… and become complicated for example an old Cadillic vs a Toyota Camry, the camry may show more damage, the occupants of the cadillic more hurt though. Camry has better crumple zone designs, which allow the front end to become damaged, but that creates a concertinaed pillow of metal which protects the occupants…
Not sure what you’re trying to say here, because for all intents and purposes, two identical cars hitting head on at 60MPH is the same as one car hitting an immovable wall at 60MPH. The moving wall example you you gave is not relevant to the discussion since the point of the immovable wall in the example is that all energy is being absorbed by the car.
While it may not be identical, it’s the same amount of force being disipated over the same distance of crumple. It’s pretty much exactly the same thing.
It’s not just the crumple zones. If you hit a car out in the open vs. hitting a car that’s already snug up against a brick wall, both at the same speed, the wall hit will still be worse, even though both have the same crumple zones involved. The mass is also relevant.
And one way in which all of these theoretical results do differ significantly from the real world is that cars are not symmetric. Most notably, they have a steering column on one side but not on the other. When two cars hit exactly head-on, the steering columns can pass beside each other, while that’s not possible with the brick wall. Now, in modern cars, the steering column is designed to collapse very easily, because it was indeed a significant danger in older cars, but I expect there’s still some effect from it.
Hey maybe I missed it among the posts but a car at 120mph hitting an identical stationary car would result in them both continuing at around 85mph according to me, NOT 60mph as I saw mentioned a couple of times. There’s just not energy in two cars moving at half the speed compared to one car moving at double the speed.
My brain sees it like this…
120mph squared = 120 X 120 = 14400 ‘units’ of energy in the moving car.
After the collision you claim 2 cars at 60mph so we have 2 x 60mph squared = 2 X 60 X 60 = 7200 ‘units’ of energy only half of what we need.
I say it’s 2 cars x 84mph squared = 2 X 85 X 85 = 14450 ‘units’ of energy which (the slight excess because I rounded the speed to 85mph) is what the original 120mph car had.
Who’s right?
And for clarity to my post above, the 85mph assumes the stationary car is in neutral with no brakes.