In your time motoring forth the roads, highways, and under-surveilled capital aliment courts of America, it’s not absurd that you’ve apparent a authentic window sticker on certain, let’s say, added sedate vehicles. The sticker, advised to blow some self-deprecating fun, states that the car that bears it can advance from 0-60 in 2.3 seconds, but afresh qualifies that with the parenthetical, off a cliff. The catechism actuality is: is this sticker true?
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Most often, you’ll see these stickers on Priuses or conceivably old Volkswagen buses, or added cars with agnate reputations for actuality slow. I get the sentiment, and acclaim the alertness to acquire and beam at one’s own car, but I’m not assertive that any car, back alone off a cliff, absolutely would hit 60 MPH in 2.3 seconds. So let’s amount out absolutely what’s activity on, application a little article I like to alarm “math.”
I’m activity to use the squishy, clammy math-brain genitalia of our own David Tracy to advice me with some of this, which I appetite to accomplish bright afore anyone all-overs to the cessation that I’m not an idiot. Aloof actuality up advanced here.
Okay, first, here’s what we know: bold this sticker’s ambition admirers is the bodies of Earth, we apperceive that gravitational dispatch on a accustomed object, like a Prius, is about 9.81 meters per second^2, or about 32.15 anxiety per second^2.
Now, if we avoid actual important factors like air resistance, we can compute that to ability 60 MPH (or 88 anxiety per second), we’ll charge to advance for about 2.74 abnormal (that’s aloof your basal “final dispatch equals antecedent dispatch additional dispatch times time kinematic equation).
So, appropriate there, we can say that this sticker is not accurate.
THIS COMICAL STICKER IS LYING TO US ALL.
In fact, to hit 60 MPH in 2.3 seconds, the boilerplate dispatch of force would charge to be 11.65 m/s^2.
So, at atomic on Earth, that sticker is a lie.
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So, okay, it’s a lie, but why stop there? Let’s amount out what an accurate, self-deprecating sticker suggesting that a driver’s own car should collapse off a bluff should be.
To do so, we charge to accomplish what David told me is alleged a chargeless anatomy diagram. That’s one that shows the article that’s accelerating (the car) and the armament acting on it.
In this case, the bottomward force is the accumulation of the car x gravity, and the advancement force is 1/2*cdA*v^2*air density, or, as David bankrupt it down:
1. Aboveboard area
2. Annoyance Coefficient
3. The aboveboard of the dispatch of the advancing air
4. Anatomy of the advancing air
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The annoyance accessory times the aboveboard breadth of a 2010-2015 Toyota Prius as apparent with that sticker on that Imgur articulation can be begin online. That Prius has a Cd of 0.25, assorted by a aboveboard breadth of 23.4, giving a CdA of about 5.85 (though it says 5.84 here). In metric, the aboveboard breadth is 2.17 m^2, consistent in a CdA of 0.54 m^2.
Okay, now let’s accomplish David do a lot of algebraic for us.
So, to get started, aboriginal David drew his chargeless anatomy diagram, summed all the forces, and set them according to accumulation times acceleration. Next, he got an blueprint with a (acceleration) and t (time) in it.
Here’s what that looked like on paper:
And, back we accept cyberbanking accuracy to advice us with this stuff, he acclimated some of those accoutrement to access at that equation.
Next, David fabricated a spreadsheet that acclimated these equations and stepped up time from 0 in 0.05 additional increments, until dispatch equaled 60 mph (28.6 m/s).
As David explained to me,
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To get dispatch at any accustomed time (for example, time = 0.15 seconds), I artlessly averaged the dispatch at 0.15 abnormal with the dispatch at the antecedent time footfall of 0.10 seconds, and assorted that boilerplate dispatch during that amount by how abundant time delayed (i.e. 0.05 seconds). That’s the change in dispatch during that time, so to acquisition the absolute velocity, you add that change in dispatch to the dispatch at the antecedent time step.
In the end, David begin that it would booty about 2.8 abnormal to hit 60 mph.
Because David is a loon, he wasn’t absolutely annoyed with the algebraic of that method, and approved again, because
The antecedent adjustment didn’t assemble aloft 202 m/s (about 450 mph) as time went to infinity. (That terminal dispatch is article we apperceive based on the annoyance coefficient, air anatomy and agent mass). Aloft additional reflection, that adjustment doesn’t accomplish a lot of sense.
Okay, fair point. So, this time, David did it the appropriate way, and set up a cogwheel blueprint (dv/dt = g-(0.3323v^2)/m), ultimately analytic for dispatch in agreement of time. His blueprint ends up being: v(t) = 201.79*tahh(0.0486*t).
Plugging in 60 mph (26.8224 m/s), our affable adjacency WolframAlpha spits out the badly altered result: 2.75 seconds. Abundant better.
Also, we now accept a admirable blueprint of the dispatch vs. time of a Toyota Prius in chargeless fall. Notice how it converges aloft the 202 m/s terminal dispatch (452 mph), which we knew as the aboveboard basis of 2*weight disconnected by the artefact (drag coefficient*density of air*frontal area).
So, attending at that! I’d say we’ve proven, above any reasonable doubt, that that sticker is asleep wrong. A car like a Toyota Prius, back alone off of a cliff, will not ability 60 mph in 2.3 seconds. It will ability 60 mph in 2.75 seconds.
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Please acclimatize your absurd stickers accordingly.
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