The Tappet Challenge

It seems my brother Nate is becoming a major source of blog material.  The other day, he pointed me to NPR's podcast of last Saturday's Car Talk.  Near the end of the show (44:38, if you'd like to hear for yourself) the hosts take a call from a woman looking to prove her husband wrong about his driving style.  The husband claims that you can get better gas mileage by accelerating up to some top speed, then allowing the car to coast until it slows to a minimum speed, before accelerating once again to the top speed.  Click and Clack maintain that, whether or not it really is better for your mileage, any money you save will be outweighed by the damage you do to your car.  They invite (or perhaps dread) any physicists listening to send them evidence one way or the other, so here I am.

We'll assume there are only two significant forces on the car: the wind resistance slowing it down, and the car's own acceleration speeding it up.  Given this, we can write a differential equation for the car's position:

where a is the acceleration applied by the car, and c is a constant associated with the wind resistance.  Solving the equation gives

which is admittedly pretty awful.  However, we're not really interested in t, so we can differentiate this to get the velocity, and solve the two equations simultaneously to get

We'd like to know the distance traveled to get from an initial speed vi to a final speed vf.  We can find this by taking the difference between the two points:

Now that we have this equation, we can find the distance the car travels during one cycle of accelerating and decelerating.  If we call the car's minimum and maximum speeds vlo and vhi, then the distance spent accelerating is

and the distance decelerating is

Using these, we can define an efficiency, ε, analogous to a measure like miles per gallon:

where E is the energy expended in traveling the distance.  It will be given by

where m is the mass of the car.  Plugging everything in, we have

where

We'd like to compare this to the efficiency of maintaining a constant velocity.  First, we need to know the average speed of the accelerating car.  Going back to our original differential equation, we can get the time in terms of the change in velocity:

We can use this to find the average velocity with

but the result is pretty awful, so I won't write it out just yet.  In this case, our efficiency is given by

so that makes our nasty equation even worse.  To spare you the horror, I decided to toss in some dummy values and see what sort of results we get.  To get a comparison of the two efficiencies, I divided the constant speed efficiency by the accelerating efficiency, so any values greater than 1 indicate that constant speed is the way to go.

Using the 65/55 speed range they discuss in the show, we get

Acceleration	Ratio
1	1.00697
3	1.00697
5	1.00696
7	1.00696
9	1.00696
11	1.00695
13	1.00694
15	1.00693
17	1.00692
19	1.00691
21	1.00688
23	1.00685
25	1.00679

so it seems Click and Clack were correct in their theory that the husband's method does not save gas.  I tried fiddling around with the speeds a bit, and I couldn't find a case where accelerating and decelerating is a good idea.  However, these equations I'm using are a real mess, so I can't be sure I've got everything exactly right.

Thanks for another great tip, Nate! I considered closing with, "Don't drive like my brother," but I don't even have a license...

Magneto Goes Back to School

Nate asked me a couple questions about my last post, which I thought I would take a minute to answer.

Can you give a more detailed explanation of magnetic fields with a non-zero second derivative? Can we generate them in a lab? Do they occur naturally?
Yeah, I glossed over that a bit.  The idea is that the field can't just be changing – the rate of change must change.  As complicated as that sounds, it's actually really easy to do.  If you hold a magnet in your hand and move it around, you're making a magnetic field with non-zero second derivative.  However, if you wanted to generate even visible light (let alone the deadly ionizing radiation I was talking about) you'd need to move the magnet up and down on the order of quadrillions of times per second.  Who knows how quickly Magneto can change his fields, but maybe with a bit of practice he could manage it.

If I have an electromagnet and I vary the input power, would that produce photons?
This may seen like a more reasonable way to get the fast variation that I'm talking about, but now you'll run into the problem of inductance.  Electromagnets create magnetic fields from electric current, but when you increase the current in an electromagnet, the magnetic field will also increase.  This creates an electric field which opposes the increasing current, slowing things down.  The same happens when the current decreases, so you still may not be able to get the speed you need.

And are the photons being produced, or just redirected?
I think it's fine to say that they are being produced, but this is getting into an area of physics I don't know much about yet.  Photons are quantum mechanical particles, while electromagnetic fields (at least the sort I'm talking about) are classical approximations of whatever is studied in quantum field theory.  Technically, I'm not sure if it's ever correct to say that a photon is 'redirected,' but they can be absorbed and re-emitted in a different direction.

Thanks for more great questions, Nate!  Everyone else should feel free to send me their own – part of the purpose of this blog is to educate, so don't feel afraid to ask even the simplest questions.

Magneto Gets an F

Last night I watched X-Men: First Class, and enjoyed it very much; it was far better than I remember the earlier movies being.  However, there's something that has always bothered me about the X-Men franchise: it seems Magneto, the mutant who can create powerful magnetic fields at will, never took physics.  There's much more he could be doing with his power, aside from tossing metal things around.

Before I get into that, I'd like to figure out exactly how powerful his magnetic fields are.  During one scene, he lifts a submarine out of the water and into the air:

A little research suggests that, although this is not a real submarine model, the Type XXI is a good approximation.  Using the length of the sub as a scale, we can estimate that the center of the sub (where we'll assume its center of mass is) is about 21 meters above the bottom of the picture.  The energy required to lift something to a specific height is given by

where m is the mass of the object, g is the acceleration due to gravity, and h is the height to which it is raised.  We can get an underestimate of this energy by assuming the submarine was on the surface of the water when Magneto started lifting it.  Plugging in the values we have, it would take 3.3 x 10^8 Joules.  I estimate it takes him about 20 seconds to get it there, giving a power output of 1.7 x 10^7 Watts.  [Update: I realized that, if that energy is coming from Magneto himself, he must be eating at least 80,000 Calories per day.]

With that in hand, we can turn to other uses of Magneto's power.  Maxwell's Equations tell us that changing magnetic fields induce electric fields, the simplest application of that being to create currents in conductors.  However, if he can create 'accelerating' magnetic fields (ones with non-zero second derivative), he can shoot lasers.  Light is made up of oscillating electric and magnetic fields, and since each induces the other, you only need one to get things started.  If he could manage to produce gamma rays (a dangerously high-energy variety of photons), according to our power output calculated above, he'd be putting out 8.6 x 10^20 photons/second.  That's about on the order of how many photons a lightbulb produces, but we're talking about a lightbulb putting out lethal radiation.

I suppose it wouldn't make for a very interesting story if Magneto simply gave everyone on Earth radiation poisoning, so maybe the writers are justified in ignoring this possibility.  It still would have been better than this possibility.