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  • BEN SPARKS: Let's say, you're in Britain on a dual carriageway.

  • I'm gonna try and draw this. Two lanes, and we'll both go in this direction

  • and there's a blue car on the left, and it's going at 70 miles per hour

  • which is the legal limit on motorways and dual carriageways in this country.

  • BRADY HARAN: They're doing the right thing.

  • BEN: Yeah, and so they should be.

  • Trouble is, you could be doing the right thing, and other people on the roads might not.

  • There's a guy overtaking the blue car. It's gonna be the red car.

  • BRADY: It's always the red car!

  • BEN: It's always the red car, and he's going faster.

  • In fact, I'm gonna draw him in at the moment he overtakes

  • and he's going -- as I'm sure it's happened to me, in the blue car I hasten to add

  • I've been in the blue car, and someone comes barreling past me going at 100 miles an hour.

  • Certainly it could happen, whether it's on a motorway or dual carriageway.

  • Now, they're not going the same speed, but there is a moment when they're neck-and-neck

  • and that's the moment I want to freeze time and ask you about what happens next.

  • Because at this moment when they're neck-and-neck, they see in front of them a tree

  • or some barrier which is unexpected, and so they both need to break

  • and we're going to assume lots of things, like is always the case with a mathematical question.

  • We have to model it to keep it simple enough to answer.

  • So the models here might be that, let's assume have same cars.

  • Let's assume reaction times are not part of this.

  • We could add those things in later if we want to complicate the model.

  • Let's assume they both slam on the brakes at the same time, and they break at the same rate

  • and by the time he gets there, the blue car has sort of just stopped in time.

  • So that's the blue car, who's got smaller apparently...

  • He has stopped with millimeters to spare, which is good.

  • BRADY: So the blue car just, just stops as it gets to the tree.

  • BEN: No crash for the blue guy. He slammed on the brakes just in time.

  • What I'm gonna claim, it's obvious that the red guy is in trouble.

  • The red guy is moving quicker. He brakes at the same time, at the same rate.

  • I think he's definitely gonna crash. No one seems to argue about whether the red guy's gonna crash.

  • What's really crucial, and here's the puzzle for you

  • is that, it matters what speed you crash at.

  • Right, if you crash at one mile an hour, no one's gonna really worry.

  • If you crash at a thousand miles an hour, you're in trouble.

  • Now, he's not gonna crash at thousand miles an hour. He's gonna crash slower because he started braking.

  • But, at what speed does he hit the tree? And that's the puzzle.

  • There's some intuition to get involved here, there's also some maths to check it out

  • and the answer is-- well, it surprised me.

  • BRADY: # Do do do do-do do do #

  • # The girl from Ipan-- # (BEN laughs)

  • BRADY: Alright. BEN: We're back! So the red guy has crashed.

  • I just feel like I missed my chance to draw a crash.

  • So this is the red guy.

  • He's in pieces all over the place.

  • Because your intuition doesn't work well with speed questions

  • and this is why I got a bit surprised about this.

  • The first thing to notice is that, a lot of people think the answer is 30 miles an hour.

  • It's really 'sensible' to think about the difference in speed, which is 30 miles an hour at the start

  • and to think, if everything else is the same, surely that's the speed it's got left over.

  • There are lots of reasons why it's not correct, and one of the reasons you'll probably think is

  • I wouldn't be asking this question if it was 30, fine!

  • Another intuition thing to ask though, is

  • do you think it's slower than 30, or do you think it's quicker than 30?

  • Because if it's not 30, it's one of the two.

  • And I'm gonna tell you the answer, and I'm gonna justify it,

  • because when someone told me the answer, I just bawled. It was...

  • I was surprised. I was also a bit shocked and dare I say it, sobered

  • and maybe I changed my driving habits a little bit, because

  • the red guy hits the tree

  • assuming all that stuff we assumed, at approximately

  • 71 miles per hour.

  • BRADY: Seventy-one! BEN: Seventy-one.

  • He has not even reached the speed that the blue guy started braking at

  • and he's hit the tree at that speed.

  • There's the difference between not crashing at all, well done blue guy

  • and hitting a brick wall, or a tree in this case, at 71 miles an hour.

  • And that was from a difference of 30 miles an hour to start with, and everything else is the same.

  • It's pretty easy to justify this, but

  • the answer is sobering enough.

  • So I think we should justify it before I leave you

  • to go and change your driving habits, and myself on my way home.

  • There are many ways to solve this

  • and maybe if you've solved it a certain way, try solve it another way.

  • You can use things called the SUVAT equations.

  • I've heard people make up nice rhymes to remember these things.

  • They're to do with constant acceleration equations to do with speed, distance, and time.

  • But there's another way which gets you the answer more quickly.

  • So maybe you should go and try this SUVAT thing

  • but if you think about kinetic energy

  • this actually is quite easy.

  • So I'm gonna imagine that they

  • I'm gonna do lots of slightly dodgy things here

  • but you have to imagine that I'm just getting a ballpark for this model.

  • The kinetic energy for the blue guy is gonna be to do with several things.

  • There's a formula for kinetic energy, which looks like this.

  • There's a half in there for reasons I'm not going to go into

  • there's the mass of the car which we're going to assume is the same for both cars

  • and there is its velocity.

  • Now, if we're gonna actually work out the kinetic energy

  • we'd work this out in meters per second, because that's the units that kinetic energy works in.

  • I'm just gonna dodge that. So I'm actually gonna say that it's kind of

  • kinetic energy is proportional to this, because I'm just gonna work with 70's.

  • But I hope you forgive me for this dodge

  • because I'm gonna do the same with the red car so we can compare them.

  • So the blue guy has got ½m, whatever the m is, and we've got 70 squared.

  • That's its velocity in miles per hour, and we won't worry about it being in meter per second.

  • Red guy, easy enough calculation

  • has a kinetic energy at the start of a ½m.

  • We're going to see that's the same mass because it's the same car

  • but this time we've got a 100 squared, because that's his velocity to start with.

  • And I'm gonna claim that in braking, you're getting rid of kinetic energy.

  • I don't think that's controversial

  • and the blue guy's got rid of all of it, so there's none left.

  • So what we need to work out is how much energy that is

  • and if actually I calculate -- the only number in here is 70 squared, and if the

  • so I'm gonna say, this is proportional to 70 squared, which is

  • 4900 amount of energy.

  • It's not in joules because I'd need to have the units changed.

  • But the red guy is proportional to 100 squared

  • which is 10,000.

  • So even if the red guy gets rid of all the energy the blue guy's got rid off

  • which he would if everything else is the same

  • I just need to do 10,000 take away 4900

  • which is 5100 of whatever units we're using left

  • which is more than the blue guy started with

  • and if you square root this, you get 71 point something

  • and that's it. The red guy's doomed

  • and it gets worse, because we haven't even considered what'd happen if they

  • maybe take a tiny bit of time to react to the tree.

  • If they take the same time to do it, the blue guy has traveled less far than the red guy

  • so it's looking even worse.

  • One other thing to bear in mind is maybe

  • you brake quicker when you're moving quicker because there's more friction

  • but that doesn't counteract the fact that the red guy is now doomed.

  • The first time I heard this story,

  • I had to go on

  • a course

  • about speed, being aware of speed on the roads.

  • BRADY: Why were you doing that course? BEN: I had, well I... it was a good idea.

  • I think it was a good idea for me to do that and avoid a massive fine.

  • But they were telling me all this stuff about driving slowly is important

  • and I'd just heard this, and this completely sobered me up

  • and they were telling me a lot of other facts, which had no effect, so

  • I tried to tell them this, they didn't appreciate that.

  • But it is powerful. It made me slow down

  • and the crucial thing to take away is that, it's the square of your speed that affects all your energy

  • and that's the bit our intuition is really bad at.

  • I'm gonna patent this one day. No stealing, Brady.

  • If speedometers in a car, instead of going 10, 20, 30, 40, 50...

  • if they went up with gaps that were proportional to the square of those numbers

  • then you'd get this lovely intuition that going from 70 to 100 is like doubling your energy

  • instead of just going up to 30 miles an hour.

  • Car manufacturers haven't taken this one on board yet

  • but, one day.

  • BRADY: I hope one of the things this video shows is that

  • in mathematics, sometimes rather than just plugging numbers into equations

  • it's good to really understand what's going on under the hood.

  • See what I did there? "Under the hood."

  • Wrote that myself.

  • But seriously, the quizzes and puzzles at Brilliant.org give you these insights.

  • They really help you see how everything fits together. They encourage critical thinking.

  • Don't just think "what formula do I use," but understand the concepts that are at play.

  • Go check out Brilliant.org/Numberphile and get 20% off Brilliant's premium membership.

  • The first 71 people to do it will be eligible for that discount.

  • And Brilliant doesn't just cover math.

  • They have all sorts of cool physics, astronomy, computer science, logic.

  • That's 20% off, at Brilliant.org/Numberphile

  • and our thanks to them for supporting this video.

  • Alright, let's really speed things up here!

  • I know I'm not supposed to be encouraging speeding

  • but it's okay a racetrack, isn't it?

  • This is a controlled environment.

  • Oh! Crashed...

BEN SPARKS: Let's say, you're in Britain on a dual carriageway.

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A2 kinetic energy kinetic brady speed energy hour

Calculating a Car Crash - Numberphile

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    林宜悉 posted on 2020/04/13
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