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Static friction is something sticky (as is Scholarship physics) Marcus Wilson Feb 13

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In January I had a go at the 2014 Scholarship Physics Exam, as I've done for the last couple of years. Sam Hight from the PhysicsLounge came along to help (or was it laugh?) The idea of this collaboration is that I get filmed attempting to do the Scholarship paper for the first time. This means, unlike some of the beautifully explained answers you can find on YouTube, you get my thoughts as I think about the question and how to answer it. Our hope is that this captures some of the underlying thinking behind the answers – e.g. how do you know you're supposed to start this way rather than that way? What are the key bits of information that I recognize are going to be important – and why do I recognize them as such? So the videos (to be put up on PhysicsLounge) will demonstrate how I go about solving a physics problem (or, in some cases, making a mess of a physics problem), rather than providing model answers, which you can find elsewhere. We hope this is helpful. 

One of the questions for 2014 concerned friction. This is a slippery little concept. Make that a sticky little concept. We all have a good idea of what it is and does, but how do you characterize it? It's not completely straightforward, but a very common model is captured by the equation f=mu N, where f is the frictional force on an object (e.g. my coffee mug on my desk), N is the normal force on the object due to whatever its resting on, and mu (a greek letter), is a proportionality constant called the coefficient of friction. 

What we see here is that if the normal force increases, so does the frictional force, in proportion to the normal force. In the case of my coffee mug on a flat desk*, that means that if I increase the weight of the mug by putting coffee in it, the normal force of the desk holding it up against gravity will also increase, and so will the frictional force, in proportion.

Or, at least, that's true if the cup is moving. Here we can be more specific and say that the constant mu is called 'the coefficient of kinetic friction': kinetic implying movement.  But what happens when the cup is stationary? Here it gets a bit harder. The equation f=mu N gets modified a bit: f < mu N. In other words, the maximum frictional force on a static object is mu N. Now mu is the 'coefficient of static friction'. Another way of looking at that is that if the frictional force required to keep an object stationary is bigger than mu N, then the object will not remain stationary. So in a static problem (nothing moving) this equation actually doesn't help you at all. If I tip my desk up so that it slopes, but not enough for my coffee mug to slide downwards, the magnitude force of friction acting on the mug due to the desk is determined by the component of gravity down the slope. The greater the slope, the greater the frictional force. If I keep tipping up the desk, eventually, the frictional force needed to hold the cup there exceeds mu N, and off slides the cup. 

What this means is that we when faced with friction questions, we do have to think about whether we have a static or kinetic case. Watch the videos (Q4) you'll see how I forget this fact (I blame it on a poorly written question – that's my excuse anyway!). 

 

*N.B. I have just picked up a new pair of glasses, and consequently previously flat surfaces such as my desk have now become curved, and gravity fails to act downwards. I expect this local anomoly to sort itself out over the weekend. 

P.S. 17 February 2015. Sam now has the videos uploaded on physicslounge   www.physicslounge.org  

Modes of a square plate Marcus Wilson Jan 15

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Alison has drawn my attention to this video. It demonstrates vibrational modes of a square plate by using sand. At certain frequencies, there are well defined modes of oscillation, in which parts of the plate 'nodal lines' are stationary. The sand will find its way to these parts and trace out some lovely pictures. 

Vibrational modes are often illustrated through waves on a guitar string. Here, the string is held stationary at both ends, but is free to vibrate elsewhere. There is a fundamental frequency of oscillation, where the distance between the ends of the string is half of a wavelength (this ensures the displacement of both ends of the string is zero since they are clamped).  Since wavelength is related to frequency (frequency = speed/wavelength) that means if the wavelength is 2 L where L is the distance between the ends of the string, we have frequency = speed/2L.  

But that's not the only possible mode. Another one would have L equal to a whole wavelength (equals two half wavelengths). Or one-and-a-half wavelengths (equals three half-wavelengths.) This gives us, rather neatly, frequency = n speed/2L, where n is an integer. We see that our 'harmonics' are just integer multiples of the fundamental frequency. Rather neat.

However, if you look at the frequencies given in the video, they appear to be all over the place. I challenge you to pull out the relationships between these (I've tried). There are a few reasons why the case shown on the video is considerably more complicated than the waves on the string. 

1. The boundary conditions. The edges of the plate aren't clamped in place. This makes it less straightforward to define the modes geometrically. 

2. The plate is square, giving rise to 'degeneracy' in the modes. This term refers to two or more distinct modes having the same frequency. You can see it rather well with the 4129 Hz mode. Basically, there are horizontal stripes shown. But equally, with the same frequency, you could get vertical stripes. Why don't the two occur together? They do. You can see the effect of having a little bit of vertical stripe most clearly at the far end of the plate, where the pattern becomes more square-like. Also, with a square, you can get two completely different types of mode with the same frequency. This occurs because what matters are the sums squares of pairs of integers. Broadly speaking (at least for a square clamped on the edges, which I must point out this ISN"T), our modes follow the relationship:

f = C sqrt(n^2 + m^2)

where C is a constant, 'sqrt' means square-root, and n^2 is n-squared. So, for example, not only is 50 equal to 5-squared plus 5-squared, it is also equal to 1-squared plus 7-squared (or 7-squared plus 1-squared). This gives us three  modes all competing to appear at exactly the same time. What happens then isn't easy to tell. 

3. Non-linear effects. This a physicist's code-word for 'it's all too difficult'. That's not quite true – arguably most of the interesting physics research happening in the world is looking at non-linear effects. What this really means is that, if A and B are both solutions of a problem, then some combination of A and B is NOT a solution. A lot of physics IS linear – Maxwell's equations in a vacuum is a good example – but a whole lot isn't. With waves, the speed of the wave usually depends on frequency (i.e. is not constant) which means we lose the nice, integer-multiple relationship of our waves-on-a-string mode.

So, enjoy the video for what it is, and don't try to analyze it TOO closely. 

 

 

Archimedes principle: think carefully Marcus Wilson Nov 14

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Benjamin has recently acquired a 'new' book from Grandma and Grandad: Mr Archimedes' Bath (by Pamela Allen – here's the amazon link – the reviews are as interesting as the content). The story-line is reasonable guessable from the title. Mr Achimedes puts water into his bath, gets in, and the water overflows. What's going on? So we've been doing some copycat experiments – not by filling the bath right up and having it slosh all over the bathroom floor (Waipa District Council – you can rest easy about water usage)  but filling up rather more sensible-sized containers and dropping objects in.

Archimedes principle is actually a little more involved than simply saying that putting an object in the water will raise the water level. It says that the weight of water displaced is equal to the force of buoyancy acting on the object.  This picture summarizes it. That is, if an object of 2 kg floats, then 2 kg of water will be displaced. If an object is unable to displace enough water for this to be the case, it will sink. That still should be pretty easy to get, especially if you've done some experimenting. However, it can still be the basis of some really hard questions. I had one in my third year  physics exams at Cambridge. In our 'paper 3', as it was called then, the examiners had free reign to ask about ANYTHING that was on the core curriculum from any of our years of study – plus ANYTHING that was considered core knowledge for entry into the degree (which meant basically anything at all you were taught in physics or general science from primary school upwards). This paper was feared like anything – it was basically impossible to revise for*. 

Here is a question then, as I recall it from the exam.

An ice cube contains a coin. The ice completely surrounds the coin. The cube is floating in a container of water.  The cube melts. Does the water level rise, fall, or stay the same? 

Think carefully before answering. 

Now, the icecube melting question is one that is often banded about. A floating icecube will displace its own mass of water (so says Mr Archimedes). When it melts, this water will occupy the 'space' that is displaced by the cube. Consequently, the water level will stay the same. A practical example of this is in the estimation of sea-level rises due to global climate change. When the ice floating on the Arctic Ocean melts, it does not cause a sea-level rise, since it is already displacing its own weight. However, the icecap on Greenland will cause a sea-level rise as it melts, since it is currently not displacing any of the sea (since it is sitting on land.) 

However, that is not the question that is asked. Our icecube has a coin inside it. What difference does it make? Well, the icecube-and-the-coin will still displace its weight of water since it floats. However, when the icecube melts, the coin sinks and no longer displaces the same amount of water as it did when it was frozen into the cube. Therefore the water level falls. That's quite a subtle application of Archimedes principle. After the exam, a group of us sat arguing about it, till we collectively worked out what the right answer was (see – exams can be good learning experiences!). Unfortunately, at this point I realized my answer was wrong. Even still, I managed to get out of the degree with a first-class honours, so I couldn't have done too badly on this exam overall.

*The other question I remember from this paper is 'What is Cherenkov Radiation?' I didn't have a clue what Cherenkov radiation was when I sat the paper – I made up some waffly words and wrote them down and almost certainly received zero for the question.'  Later, one of my friends found a single, incidental sentence in a handout that was given out by our nuclear physics lecturer that identified what it was. That's how nasty this exam was. 

Toddler does physics-art Marcus Wilson Oct 29

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As we all know, a scientifically-minded toddler plus a piece of technology can lead to unexpected results. This is the result of Benjamin playing with a retractable steel tape measure at the weekend. How we came to break the case apart I don't know, but the results are pretty (the cellphone shot in poor light doesn't do justice to the artwork): 

20141027_145124.jpg

 

 

20141027_145132.jpg

I like the koru-shape made by the end. The measure has curled itself into a complicated form rather reminiscent of a protein structure, with sections of helices and straighter lengths. Although the mechanisms are different (protein structure has a lot to do with the intricaces of chemical bonding) the physical process is similar –  the structure works itself to a local minimum of energy. Just how this happens  is all rather complicated from a physics perspective. Perhaps the most obvious example of twists of this form is in telephone cords. The phenomenon has even lent its name to a type of structure seen in thin films – the 'telephone cord buckle'. Unfortunately Benjamin didn't give me any warning about what was going to happen – otherwise I'd have filmed it (and he would probably have retreated to a safe distance – the whole unravelling was pretty energetic). 

BUT…since Karen is an occupational therapist and has accumulated large numbers of free tape measures as corporate freebies in her career, we could maybe spare a few for high-speed filming.

Robot racing Marcus Wilson Oct 22

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The Engineering Design Show is currently in full swing here, with the competitions for the various design projects. The white-line followers kicked off proceedings. They were pretty impressive, with all but one team successfully being able to follow the (very squiggly) line without mistakes. There were traps to confuse the robots – the line got thinner and thicker, crossed over itself, had abrupt corners and so on, but the robots were well programmed and coped with this easily. The winning group was impressive indeed. They had some very carefully optimized control parameters, meaning that the robot was (a) really straight and fast on a straight-line section but also (b) precise round the turns, slowing down just enough to take each turn at about the right speed. I think anyone would struggle to get something going quicker than this one. 

On show at the moment are the third year mechanical engineering students who have designed a pin-collecting machine. The idea is that the vehicle pulls still pins (about 5 cm in length, maybe 5 mm in diameter) out of a board – the one that collects all the pins in the quickest possible time and drops them back in the collecting bin is the winner. The most striking conclusion from this exercise is the emphasis on the old adage "To finish first, first you must finish". A good proportion of the entries have died part way through the process – pins have jammed the mechanisms, the motors have failed, or, in one disappointing case, the machine collected the pins in lightning quick time and then failed to go back to deposit them in the collecting bin. Also, we've seen one machine disqualified for being downright dangerous – its first run saw it pulling pins out of the board and firing them across the room causing spectators to beat a hasty retreat. 

But the winner (or so it looks) has pushed their luck to the limit.  The "…first you must finish" line is actually not quite correct. More accurate would be to say "…second you must finish. First, you must start". They've admitted to putting 5 volts over a motor rated at 3 volts in practice just before the event, and frying the motor. They then had to hurridly locate a replacement and install it while the competition was in progress. Missing their first two rounds, they appeared looking hot and sweaty just in time for their run in round 3 out of 4 and simply destroyed the rest of the competition. (Presumably it won't be long before they destroy their new motor too, but it's survived long enough to win, according to the rules, and that's what counts.)

Overall the design show has been great fun to be a part of and has really demonstrated the skills that the students have acquired. Well one everyone involved!

Postscript 29 October 2014: We're a hit with the Waikato Times!

Telepathy breakthrough – great science, not science fiction Marcus Wilson Sep 08

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The 'Science' news hitting the media at the weekend was Guilio Ruffini and Alvaro Pascual-Leone's demonstration of 'telepathy'. There's been a lot of media coverage on this – for example the neat little interview of Ruffini on the BBC's 'Today' programme.

Their article on this can be read here. It's not a long one, and, for a piece of science, I reckon it's pretty clearly described. 

But, I'm afraid, you can forget The Chrysalids – the messages sent from India to France are of a rather more humble nature. But the science behind it is great. 

Essentially, the work has linked together two existing technologies, via the internet. The first is long-established – namely monitoring of the electroencephalogram (EEG). If you place electrodes on the surface of your scalp, you can detect electrical signals that originate from the electrical behaviour of the neurons in the cortex of your brain. The signals aren't large, just a few microvolts, but they are fairly easy to pick up. I get students doing it in the lab. Different kinds of brain activity lead to different signal patterns. A 'thinking' brain has lots of small amplitude, fast activity, whereas someone in deep sleep shows an EEG pattern that has a large, approximately 1 Hz cycle to it. The two patterns are very different. EEG is routinely used for monitoring sleep patterns and as a tool for an anaesthetist to monitor the depth of anaesthesia in their patient – one wants to make sure the patient is well anaesthetised, but on the other hand one doesn't want to head into Michael Jackson territory. The EEG can help. 

So the EEG is a way of 'reading' the state of the brain. To go from an EEG recording to working out what the subject is thinking about is a long, long way off, if indeed it's possible at all, but one can certainly say something about the brain state. 

If EEG is about reading the state of a brain, then the other technology, transcranial magnetic stimulation (TMS), does the reverse. This is rather newer, and our understanding of it is much poorer (I'm involved with a TMS research project at the moment).  In TMS, pulses of magnetic field are applied to the brain. The effect depends on what area of the brain the pulse is applied to, and in what orientation. At a simple level you can make an arm 'twitch' by applying the pulse to the correct part of the motor cortex. I've seen this done at the University of Otago (on a brave summer student of mine). In Ruffini's work, they used the magnetic pulse to 'create' the perception of a flash of light by stimulating the visual cortex. The subject 'sees' the light, even though there's no such flash on the retina, since the sensory circuits in the cortex that usually interpret what's going on on the retina are activated remotely. 

So what did the experiment do? The person in India sending the message imagined a particular activity (hand or foot movement), and their EEG changed depended on whether they imagined the hand or foot. A computer interpreted the EEG, decided on which it was, and communicated with the computer in France. The French hardware system then zapped the human receiver in such a way as to either trigger the flash or not trigger the flash. The receiver then reported orally whether they'd seen a flash. In this way the 'message' (a string of 1's (hands) and 0's (feet) ) has been sent from one to another without using the senses of the receiver. 

In that sense this is telepathic. The receiving person had no communication with the transmitting person in a visual, oral, or any other way. True, one might ask, why didn't they just phone/Skype/email each other to send the message, and of course you wouldn't want to communicate with your family members overseas with an EEG/TMS system. But that's not the point. The point is that it is a great demonstration of science. 

Will it lead to small telepathic headsets? Rather than fuss with phones and email, we could just have a conversation with anyone in the world just by thinking about it. (You'd want to be sure you'd switched it off afterwards!)  Don't get excited – we're not in Chrysalids territory yet. That's a long, long, long, long way off. But it is good science. 

 

 

 

Engineering, lego and line followers Marcus Wilson Aug 19

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In the last few weeks I've been working with some second-year software engineering students on a design project. Their particular task is to build (with Lego – but the high-tech variety) a robot that can follow a white line on a bench. It's fun to watch them play with different ideas and concepts – there's the occasional disaster when the robot roars off at high speed in an unexpected direction and falls off the bench top. 

To produce something that approximately works isn't that difficult. We can use a couple of lights and detectors, sitting either side of the white line. If the robot is going straight, neither gets much reflection. But if one records a high amount of reflected light (and so is on top of the line) we need to turn the robot  - if it's the other that records a high amount, we ned to turn the other way. Indeed, many, many years ago I made something very similar using analogue electronics (a few LEDs, photocells, transistors etc and a couple of motors to turn the wheels). It approximately worked, but there were a lot of conditions that would fool it – give it shadows and sharp corners to deal with and it was lousy. 

The lego robots that the students have can be programmed – and as such there is a huge array of different options for their control. The exercise is just as much in the development of the software as the hardware. Indeed, since these students are software engineering students, that is the bit they are most familiar with. 

One thing we're trying to get them to think about are different concepts. It's easy to think of one solution and just go with that. But is that the best solution? In engineering we can't afford just to develop the first idea that comes into our heads. We don't really have much idea about what is 'best' until we think through other possibilities and assess them against relevant criteria.  Too often we can be constrained by traditional thinking – "it has to be done that way" – without really considering novel options.  Two light sensors might work. But would three (or even four) be better? How are they best placed?  What about sensors that aren't rigidly mounted but can move (actively search for the line)? The possibilities are almost endless. 

But the hardware is only half the problem. How should the robot best respond to the input signals? Simply turning one way or the other is easy to implement, but can lead to excessive oscillation. There are smarter control systems available (e.g. Proportional Integral-Differential control), but at a cost of increased complexity. Is it worth pursuing them?

These are questions that the students need to think about with their project. We can get them to do that (rather than just thinking up one solution that might work and considering nothing else) by setting the assessment tasks appropriately. So they are not just judged on how well their robots can follow the white line, but what concepts they thought about, and whether they selected one appropriately using reasonable specifications and design criteria (i.e. how well they followed the established process for engineering design). In fact, following the design process well should ensure that the end result actually does do a good job of following the line accurately, repeatedly and quickly . 

There are still several weeks until the end of semester, when these line-following robots need to be perfected. They'll be tested at the Engineering Design Show where we'll find out how to build a good line-follower.

 

Going down the plughole Marcus Wilson Jul 04

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Being a father of an active, outdoor-loving two-year-old, I am well acquainted with the bath. Almost every night: fill with suitable volume of warm water, check water temperature, place two-year-old in it, retreat to safe distance. He's not the only thing that ends up wet as he carries out various vigorous experiments with fluid flow. 

One that he's just caught on to is how the water spirals down the plug-hole. Often the bath is full of little plastic fish (from a magnetic fishing game), and if one of these gets near the plug hole it gets a life of its own. It typically ends up nose-down over the hole, spinning at a great rate as it gets driven round by the exiting water. 

The physics of rotating water is a little tricky. There are two key concepts thrown in together; first the idea of circular motion  - which involves a rotating piece of water having a force on it towards the centre (centripetal force); second is viscosity – in which a piece of water can have a shear force on it due to a velocity gradient in the water. Although viscosity has quite a technical definition, colloquially, one might think of it as 'gloopiness' [Treacle is more viscous than water. The ultimate in viscosity is glass, which is actually a fluid, not a solid - the windows of very old buildings are thicker at the bottom than the top due to the fluid flow over tens or hundreds of years.] In rotational motion there's a subtle interplay between these two forces which can result in the characteristic water-down-plughole motion. 

In terms of mathematics, we can construct some equations to describe what is going on and solve them. We find, for a sample of rotating fluid, that two steady solutions are possible. 

The first solution is what you'd get if all the fluid rotated at the same angular rate – the velocity of the fluid is proportional to the radius. This is what you'd get if you put a cup of water on a turntable and rotated it – all the water rotates as if it were a solid.

The second solution has the velocity inversely proportional to the radius – so the closer the fluid is to the centre, the faster it is moving. This is like the plughole situation where a long way from the plug hole the fluid circulates slowly, but close in it rotates very quickly. Coupled with this is a steep pressure gradient – low pressure on the inside (because the water is disappearing down the hole) but higher pressure out away from the hole. Obviously this solution can't apply arbitrarily close to the rotation axis because then the velocity would be infinite. This is where vortices often occur. (Actually, Wikipedia has a nice entry and animations on this, showing the two forms of flow I've described above). 

A Couette viscometer expoits these effects as a way of measuring the viscosity of a fluid. Two coaxial cylinders are used, with fluid between them. The outer is rotated while the inner one is kept stationary, and the torque required enables us to calculate the viscosity of the liquid.

 

Dismantling the health and safety pyramid Marcus Wilson Jun 04

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A few days ago I was updating one of the lectures I do for my Experimental Physics course. I was putting in a bit more about safety and managing hazards, which are things that are associated with doing experiments for real. When I was a student, we didn't learn anything about this – my first introduction to the ideas behind hazard management came only when I joined an employer. Before then, I simply hadn't thought about the issues involved. 

One of the things that gets banded around Health and Safety discussions is Heinrich's pyramid, dating back to 1931. The basic idea of this is that accidents don't just happen out of the blue. For every fatal accident there are several non-fatal but major accidents, for every major accident there's several minor accidents, and for every minor accident there's a whole heap of incidents (things that could have been accidents if circumstances had been different). The implication is then that by addressing the minor things that crop up frequently, we make the workplace a safer place.  I've seen various versions of the pyramid on-line, but here's one:

enerpipe_img-SafetyPyramid.jpg

Diagram taken from http://www.enerpipeinc.com/HowWeDoIt/Pages/safety.aspx 

That all seems to make some sense. However, searching around for a good picture to include in my lecture notes, I came across this article by Fred Manuele:

http://www.asse.org/professionalsafety/pastissues/056/10/052_061_f2manuele_1011z.pdf

It calls into question the whole basis of this pyramid and its implications for health and safety in the workplace. Specifically, Manuele reports that:

1. No-one can trace Heinrich's original data

2. If it exists, then the extent to which 1920's and 30's data applies in today's workplace is dubious. 

3. That the pyramid idea is counter-productive to ensuring a safe working environment since it over-emphasizes the importance of minor non-compliance issus (not wearing one's lab coat) and focuses attention away from major, systemic failings in senior managment and even government regulators and legislators whence the really big events tend to come. [Think Pike River, where MBIE's own investigation points the finger at itself (in the form of its predecessor, the Ministry of Economic Development) for carrying out its regulatory function in a 'light-handed and perfunctory way'.]

There's some lovely statistics included on what a focus on reducing small incidents actually does. Here's some US figures on the reduction in accdient-related insurance claims between 1997 and 2003 (F. Manuele,  “State of the Line,” by National Council on Compensation Insurance, 2005, Boca Raton, FL):

Less than $2000: Down 37%

$2000 – $10000: Down 23%

$10k – $50k: Down 11%

above $50k: Down 7%

See the issue here? Focusing attention on small incidents and small accidents does wonders for reducing small incidents and small accidents, but very little on reducing the big accidents. That's because, as Manuele describes, they have different underlying causes. 

The paper's worth a read, and cuts at what I've been taught over several years about health and safety. One notable feature is that it actually draws from hard data, rather than myth, which is how Manuele labels Heinrich's work. 

And the consequence for my experimental physics students? I shan't be including that pyramid in their lectures.

 

 

 

The gearbox problem Marcus Wilson May 27

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At afternoon tea yesterday we were discussing a problem regarding racing slot-cars (electric toy racing cars).  A very practical problem indeed! Basically, what we want to know is how do we optimize the size of the electric motor and gear-ratio (it only has one gear) in order to achieve the best time over a given distance from a stationary start?

There's lots of issues that come in here. First, let's think about the motors. A more powerful motor gives us more torque (and more force for a given gear ratio), but comes with the cost of more mass. That means more inertia and more friction. But given that the motor is not the total weight of the car, it is logical to think that stuffing in the most powerful motor we can will do the trick. 

Electric motors have an interesting torque against rotation-rate characteristic. They provide maximum torque at zero rotation rate (zero rpm), completely unlike petrol engines. Electric motors give the best acceleration from a standing start – petrol engines need a few thousand rpm to give their best torque. As their rotation rate increases, the torque decreases, roughly linearly, until there reaches a point where they can provide no more torque. For a given gear ratio, the car therefore has a maximum speed – it's impossible to accelerate the car (on a flat surface) beyond this point. 

Now, the gear ratio. A low gear leads to a high torque at the wheels, and therefore a high force on the car and high acceleration. That sounds great, but remember that a low gear ratio means that the engine rotates faster for a given speed of the car. Since the engine has a maximum rotation rate (where torque goes to zero) that means in a low gear the car has good acceleration from a stationary start, but a lower top-speed. Will that win the race? That depends on how long the race is. It's clear (pretty much) that, to win the race over a straight, flat track, one needs the most powerful engine and a low gear (best acceleration, for a short race) or a high gear (best maximum velocity, for a long race). The length of the race matters for choosing the best gear. Think about racing a bicycle. If the race is a short distance (e.g. a BMX track), you want a good acceleration – if it's a long race (a pursuit race at a velodrome), you want to get up to a high speed and hence a huge gear.  

One can throw some equations together, make some assumptions, and analyze this mathematically. It turns out to be quite interesting and not entirely straightforward. We get a second-order differential equation in time with a solution that's quite a complicated function of the gear-ratio. If we maximize to find the 'best' gear, it turns out (from my simple analysis, anyway) that the best gear ratio grows as the square-root of the time of the race. For tiny race times, you want a tiny gear (=massive acceleration), for long race times a high gear.   If one quadruples the time of the race, the optimum gear doubles. Quite interesting, and I'd say not at all obvious. 

The next step is to relax some of the assumptions (like zero air resistance, and a flat surface) and see how that changes things. 

What it means in practice is that when you're designing your car to beat the opposition, you need to think about the time-scales for the track you're racing on. Different tracks will have different optimum gears.

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