One of my favourite examples of this pedagogically is how kinetic energy is quadratic with velocity. It's easy to tie into things people think they understand, like braking distance. Our intuitions are wrong even in shockingly simple problems.
> Would you father get hit with:
> * a 1 kg object moving at 1 m/s or
> * a 1 g object moving at 1 km/s?
1 m/s is just 3.6 km/h, or slow walking speed. You'll be fine.
And to illustrate the point a bit further, that same argument doesn't work in other speed/mass regimes. Would you rather get hit with:
- A 1000 kg object moving at 1 mm/s or
- A 1 kg object moving at 1 m/s
It's suddenly no longer intuitively obvious because both are totally fine. Like...roll a die, and my squishy human body will be equally happy with either result. Or, shifting the argument the other direction, would you rather get hit with:
- A 1000 kg object moving at 1 km/s or
- A 1 kg object moving at 1000 km/s
Pretty obviously the 1 kg object, despite its much higher kinetic energy. If I'm lucky it just takes off an arm or something, whereas the 1000 kg object is definitely splattering me.
And that sort of thing is why kinetic energy can be hard to reason about. Our intuition is calibrated around our squishiness, which only loosely relates to energy. There are other factors at play.
Why?
- Why more dangerous: A 1g steel ball is around 1/4 inch in diameter. A 1/4 inch hole in any major artery is an immediate life-threatening event, same if it hits the lungs, heart, or brain (gut-reaction: I think it'd be quite hard to penetrate the skull with this projectile, so to hit the brain you'd have to go through the eyes or pterion or something, but ignoring that sort of potential quibbling). It'll have pretty significant negative effects on any other system if it can actually pierce that system. If we pick a denser material like lead, the hole is smaller but still life-threatening. If we pick a less-dense material like wood, the hole is larger, and you're still pretty majorly screwed. Contrast that with the 1kg object, which is roughly equivalent to dropping two cans of beans from a height of 2 inches; it just doesn't matter.
- Why pierce: That's a big can of worms with a lot of answers. A halfway okay way to think about it for common hard materials and common shapes in the speed regime being discussed is that you have a higher energy to surface-area ratio.
Critically (in case you're using some sort of Socratic method to guide me to the increased energy being the rationale), the important number for piercing injuries is closer to being that ratio than the total energy, hence my example of 1mg at 1km/s in some recent comment. That has the same energy as our 1kg at 1m/s object, but it's still extremely dangerous because it physically punches out a 1mm hole through your body. If that strikes a location like your eye or a superficially exposed major artery (like the one on the inside of your elbow), you're still going to have a very bad day.
Your stupid brain thinks, oh I am just 10ish over the limit.
The exact numbers are more nuanced (both in simulation and in terms of available data on the topic), but to illustrate the effect, suppose typical reaction time allows you to slow to 0mph from 25mph just before striking a child with your car, assuming somebody with perfect training and a typical car applying proper emergency braking and decelerating at around 0.8 gs. I'll consider both the 0.5s reaction time estimate (median results in places where people are primed to have to react) or 1.5s (median for an alert driver expecting hazards occasionally). Let's assume we're going 10 over.
The fact that you were going faster means you cover more distance before you even hit the brakes. The 25mph traversed either 44.43ft in 1.92s (at 0.5s reaction time) or 81.1ft in 2.92s (at 1.5s reaction time). By the time the 35mph even applies the brakes, it only has either 18.77ft left or 4.1ft left (0.5s vs 1.5s reaction time).
Then how much can you slow down in that time? In the 0.5s reaction time case the impact speed is 27.85mph. In the 1.5s case it's 33.57mph.
In either case, you're actually striking the child above the posted speed limit in the same emergency-braking scenario, at a speed likely to cause significant injury.
Details obviously vary depending on how you set up the parameters, but 25mph is an approximate cutoff below which we empirically tend to have pretty safe traffic outcomes in residential areas and above which we tend to have pretty dangerous traffic outcomes. I don't know if people choose to speed in those situations because they don't know or because they don't care, but either way, it makes our roads much more dangerous than they need to be.
Instead of mph/kph the speed limit and speedometers could be on QuadraticSpeedUnits.