In your time motoring along the roads, highways, and under-surveilled mall food courts of America, it’s not unlikely that you’ve seen a particular window sticker on certain, let’s say, more sedate vehicles. The sticker, designed to poke some self-deprecating fun, states that the car that bears it can accelerate from 0-60 in 2.3 seconds, but then qualifies that with the parenthetical, off a cliff. The question here is: is this sticker true?
Most often, you’ll see these stickers on Priuses or perhaps old Volkswagen buses, or other cars with similar reputations for being slow. I get the sentiment, and applaud the willingness to accept and laugh at one’s own car, but I’m not certain that any car, when dropped off a cliff, actually would hit 60 MPH in 2.3 seconds. So let’s figure out exactly what’s going on, using a little something I like to call “math.”
I’m going to use the squishy, moist math-brain parts of our own David Tracy to help me with some of this, which I want to make clear before anyone jumps to the conclusion that I’m not an idiot. Just being up front here.
Okay, first, here’s what we know: assuming this sticker’s target audience is the people of Earth, we know that gravitational acceleration on a given object, like a Prius, is about 9.81 meters per second^2, or about 32.15 feet per second^2.
Now, if we ignore very important factors like air resistance, we can compute that to reach 60 MPH (or 88 feet per second), we’ll need to accelerate for about 2.74 seconds (that’s just your basic “final velocity equals initial velocity plus acceleration times time kinematic equation).
So, right 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 average acceleration of gravity would need to be 11.65 m/s^2.
So, at least on Earth, that sticker is a lie.
So, okay, it’s a lie, but why stop there? Let’s figure out what an accurate, self-deprecating sticker suggesting that a driver’s own car should plummet off a cliff should be.
To do so, we need to make what David told me is called a free body diagram. That’s one that shows the object that’s accelerating (the car) and the forces acting on it.
In this case, the downward force is the mass of the car x gravity, and the upward force is 1/2*cdA*v^2*air density, or, as David broke it down:
1. Frontal area
2. Drag Coefficient
3. The square of the velocity of the oncoming air
4. Density of the oncoming air
The drag coefficient times the frontal area of a 2010-2015 Toyota Prius as seen with that sticker on that Imgur link can be found online. That Prius has a Cd of 0.25, multiplied by a frontal area of 23.4, giving a CdA of about 5.85 (though it says 5.84 here). In metric, the frontal area is 2.17 m^2, resulting in a CdA of 0.54 m^2.
Okay, now let’s make David do a lot of math for us.
So, to get started, first David drew his free body diagram, summed all the forces, and set them equal to mass times acceleration. Next, he got an equation with a (acceleration) and t (time) in it.
Here’s what that looked like on paper:
And, since we have electronic brains to help us with this stuff, he used some of those tools to arrive at that equation.
Next, David made a spreadsheet that used these equations and stepped up time from 0 in 0.05 second increments, until velocity equaled 60 mph (28.6 m/s).
As David explained to me,
To get velocity at any given time (for example, time = 0.15 seconds), I simply averaged the acceleration at 0.15 seconds with the acceleration at the previous time step of 0.10 seconds, and multiplied that average acceleration during that span by how much time elapsed (i.e. 0.05 seconds). That’s the change in velocity during that time, so to find the total velocity, you add that change in velocity to the velocity at the previous time step.
In the end, David found that it would take about 2.8 seconds to hit 60 mph.
Because David is a loon, he wasn’t entirely satisfied with the math of that method, and tried again, because
The previous method didn’t converge upon 202 m/s (about 450 mph) as time went to infinity. (That terminal velocity is something we know based on the drag coefficient, air density and vehicle mass). Upon second reflection, that method doesn’t make a lot of sense.
Okay, fair point. So, this time, David did it the right way, and set up a differential equation (dv/dt = g-(0.3323v^2)/m), ultimately solving for velocity in terms of time. His equation ends up being: v(t) = 201.79*tahh(0.0486*t).
Plugging in 60 mph (26.8224 m/s), our friendly neighborhood WolframAlpha spits out the dramatically different result: 2.75 seconds. Much better.
Also, we now have a lovely graph of the velocity vs. time of a Toyota Prius in free fall. Notice how it converges upon the 202 m/s terminal velocity (452 mph), which we knew as the square root of 2*weight divided by the product (drag coefficient*density of air*frontal area).
So, look at that! I’d say we’ve proven, beyond any reasonable doubt, that that sticker is dead wrong. A car like a Toyota Prius, when dropped off of a cliff, will not reach 60 mph in 2.3 seconds. It will reach 60 mph in 2.75 seconds.
Please adjust your comical stickers accordingly.