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Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. The only way to answer questions like this is to rely on symmetry principles, since these are more fundamental than the laws of motion. Further, if you smack two identical clay balls of mass $m$ moving with velocity $v$ head-on into each other, both balls stop, by symmetry. But now look at this in a train which is moving along with one of the balls before the collision.
The kinetic energy of the second ball is $mE(2v)$ at the start, and after the collision, you have $2mE(v)$ kinetic energy stored in the combined ball. Here is the non-circular version of the force-times-distance argument that everyone seems to love so much, but is never done correctly. That the energy in a constant gravitational field is proportional to the height is established by statics.
Another way of saying the same thing uses an elevator, consisting of two platforms connected by a chain through a pulley, so that when one goes up, the other goes down.
This establishes that for all reversible motions of the elevator, the ones that do not require you to do any work (in both the colloquial sense and the physics sense--- the two notions coincide here), the mass-times-height summed over all the objects is conserved. So the mass-times-height is a measure of the effort required to lift something, and it is a conserved quantity in statics. This is the original demonstration of the laws of elastic collisions by Christian Huygens, who argued that if you drop two masses on pendulums, and let them collide, their center of mass has to go up to the same height, if you catch the balls at their maximum point. For completeness, the fact that an object accelerates in a constant gravitational field with uniform acceleration is a consequence of Galilean invariance, and the assumption that a gravitational field is frame invariant to uniform motions up and down with a steady velocity. The question is especially relevant from a didactical point of view because one has to learn to distingish between energy (work) and momentum (quantity of motion).
The kinematic property that is proportional to $v$ is nowadays called momentum, it is the "quantity of motion" residing in a moving object, it's definition is $p:= mv$. The change of momentum is proportional to the impulse: impulse is the product of a force $F$ and the timespan $\Delta t$ it is applied. Now for an intuitive explanation that an object with double velocity has four times as much kinetic energy.
No, because it travels the second meter in a lot less time (because it's already moving), so it has less time to gain speed. The only real physical reason (which is not really a fully satisfying answer) is that $E \sim v^2$ is what experiments tell us. Of course, you could question why gravitational potential energy is proportional to height, and once that was resolved, question why some other kind of energy is proportional to something else, and so on. As Piotr suggested, accepting the definition of work $W=\mathbf{F}\cdot d\mathbf{x}$, it follows that the kinetic energy increases quadratically. So, we see that the definition of work is synonymous with quadratic dependence on the velocity. For every relatively equal (in percents) increase of the speed, the applied force must be present over increasingly (quadratically) long travel distance. In his excellent answer, Ron Maimon cleverly suggests using heat to avoid a reference to work. So, in summary, I suggest the "why" of the question is the same as the "why" of the first law. The general form of the kinetic energy includes higher order corrections due to relativity. There is another fundamental reason for which kinetic energy cannot depend linearly with the velocity. In a second set of tests the first cannon fires the spring into an immovable brick wall from a distance of one meter.
Basically, momentum is related to force times time, and KE is related to force times distance. That is, if you scale your velocity by a factor of $\lambda$, your Energy is scaled by a factor of $\lambda^2$ — this should answer your question (just plug in the numbers).
I have a quantitative answer which is a thought experiment avoiding all but the simplest equations.
Now release the spring and m1 shoots off to the left with v=2 and m3's velocity goes from v=1 to v=0 making its KE=0. I'm not happy with assuming away the mass of the spring so a neater explanation has a spring attached to each ball and the balls interact via their springs which are in contact.
The average kinetic energy of the molecules does not change with time (as long as the temperature of the gas remains constant). The pressure of a gas is causes by collisions of the molecules with the walls of the container. This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. Isaac Newton (a 17th century scientist) put forth a variety of laws which explain why objects move (or don't move) as they do. An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. But before we can start with discussion about cornering, it is important to acknowledge importance of suspensions. The racing line is the imaginary line a driver should follow to travel round corners quickly.
It is important to remember that there is rarely a perfect line through any corner for all conditions. The exit is where the car is driving straight again and we can divide this zone to acceleration and full throttle zone.
To get the best times on the track you need to be either accelerating or braking at all times while on the straights - any coasting means you're losing precious seconds!
The largest demand on the grip reserves of your tires occurs at the point of turn in up to the point of apex. For a majority of drivers front-end grip and front-end feel mean precision of corner entry and tool to carry a lot of speed into a corner and maintain it through the apex and in the same time a good feedback from the front tires. Without the response from the front end, some drivers cannot make the fine adjustments that allow them to carry speed in the most efficient way through the corner and if you cannot put the car where you want and brake where you want because of front locking or sliding, then it becomes a guessing game about where you are going to be. To carry maximum speed through a corner, you need to take the route that minimizes the corner radius, thus minimizing cornering force and frees up precious grip for keeping the speed up. Oddly enough, carrying the highest average speed round corners may not be the quickest way around a track.
If you are late with braking point, or you are tying to outbrake somebody during overtaking from inside line and you are to long in entering the corner, very important is to try to make corner entry as straight line as possible.
Once you've hit the apex, you should be able to start reducing the amount of steering lock. The position and direction of the next corner will also affect the apex, for example, if the next bend is a left hander you'll need to move over to the right hand side of the track, and thus will need to apex later. In drifting one should aim the front of the car towards the apex to initiate a high-angle drift.
Depending upon cornering situations, techniques like trail braking can be used to maintain more speed upon entry of a corner, and attaining more grip while turning into the corner, and has an effect on apex selection. It's very rare to achieve the perfect corner, it takes knowledge of the track and the car and a great deal of practice! There are many corners with famous names, such as the Parabolica at Monza, Spoon in Suzuka or Becketts at Silverstone.
The battle for the first corner is one of the most exciting moments in any racing, and especially in Formula 1 race, where overtaking during the race is very difficult. It is much easier to catch up or lose places at the start than during the rest of the race. Because jostling is inevitable in the battle for the first corner, the FIA has done a great deal to make sure it doesn't have serious consequences for the drivers or spectators. A great deal was done to improve the safety during the conversion of this high-speed track. But this is not really satisfying, because you could then ask "Why is work force dot distance?" and the mystery is the same. Using Galilean invariance, the symmetry that says that the laws of physics look the same to you on a moving train, you can explain why energy must be proportional to the mass times the velocity squared. I will define it as follows: the kinetic energy $E(m,v)$ of a ball of clay of mass $m$ moving with velocity $v$ is the amount of calories of heat that it makes when it smacks into a wall.

The result is that each acts as a wall for the other, and you must get an amount of heating equal to $2m E(v)$. In this frame of reference, the first ball starts out stopped, the second ball hits it at $2v$, and the two-ball stuck system ends up moving with velocity $v$.
If you believe the law of the lever, an object will be in equilibrium with another object on a lever when the distances are inversely proportional to the masses (there are simple geometric demonstrations of this that require nothing more than the fact that equal mass objects balance at equal center-of-mass distances). The "energy" can now be defined as that quantity of motion which is conserved when these objects are allowed to move with a non-infinitesimal velocity. From this, Huygens generalized the law of conservation of potential energy implicit in Archimedes to derive the law of conservation of square-velocity in elastic collisions. For example, gravitational potential energy on the Earth's surface is proportional to height, and if you drop an object, you can measure that the height it falls is proportional to the square of its speed. Given the homogeneity of space, it can not contain explicitly $\mathbf{r}$ and given the homogeneity of time, it can not depend on the time. Momentum grows linearly with velocity making momentum a quantity that is intuitive to understand (the more momentum the harder an object is to stop). The quadratic term is only a Newtonian approximation valid when velocities are low in comparison to the speed of light c. Upon impact the spring which has a length of 20 cm compresses to a length of 16 cm (20% compressed).
In my explanation I will use the same method to push the object from v=0 to v=1 then from v=1 to v=2, then v=2 to v=3, etc. Because we've said that the mass of the spring is very small so its KE is almost zero then all of the energy which was in the system before the spring was released is now in m1.
An object which has motion - whether it is vertical or horizontal motion - has kinetic energy.
That means that for a two time increase in speed, the kinetic energy will increase by a factor of four. If they are not set properly, or there is some malfunctioning or lack of maintenance, all this theory we will talk about are useless. If you remember one thing from Braking and advanced braking article make sure to never brake on a bend.
Engine and brakes play crucial role on the straights but on the corners, the driver's skill is major factor.
The road cars are naturally made to understeer, and the race cars are naturally made to oversteer.
The basic assumption is that a straight line between two points is usually the quickest route.
It depends on the characteristics of your car, your cornering strategy, and the conditions around you. Try to accelerate all the way up to the braking zone and use maximum throttle up to the last moment.
The golden rule here is to select a gear which will allow you to accelerate out of the bend efficiently. The perfect corner involves tightening the steering until the apex and then gradually unwinding the steering lock. It is vitally important not to place additional demands on the tires by accelerating or braking. Hitting the apex allows the car to take the straightest line and maintain the highest speed through that specific corner. There are two different types of apex, the geometrical apex (fixed or traditional), and the racing apex (fluid - early or late). In this way they can feel, without guessing, exactly what the front tires are up to, and carry the speed into the corners and keep it up in the mid-corner. And if you miss a little bit the corner you are going to miss a lot of speed on the next straight.
This route put apex near the geometric apex of the corner as is usually know as the classic racing line.
In this case, the apex for the late apex racing line is about three quarters of the way around the bend. We all know that a car which has a greater distance to accelerate will be going faster than a car which has only been accelerating for a shorter distance. High-angle drifting keeps the front of the car facing the apex, blocking the opponent from overtaking. In this technique, brake pressure is applied slightly later than usual upon deceleration, and is maintained during steering input, sometimes all the way to the apex. You will not be able to use all the power of a Bugatti Veyron or F1 car until you're completely in a straight line, with this kind of car you will have very distinguished acceleration and full throttle area.
But those are not the corners that can decide a race: on every track, the corner the drivers pay most attention to, and which commands the most interest from the fans, is the first corner after the race begins. So only one motto applies for the drivers up to the first corner: keep your eyes straight ahead and put your foot down.
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This definition does not make reference to any mechanical quantity, and it can be determined using thermometers. Then if you tilt the lever a little bit, the mass-times-height gained by 1 is equal to the mass-times-height gained by the other. You can lift two objects a certain distance in two steps, if you drop an object twice as far.
By this I mean that if you let two weights drop while suspended on a string, let them do an elastic collision, and catch the two objects when they stop moving again, you did no work. His principle that the center of mass cannot be raised by dynamic collisions is the first statement of conservation of energy. Therefore the braking distance of B will be a factor of 4 bigger then the braking distance of A (its starting velocity, and therefore also its mean velocity, being twice as much, and its time $\Delta t$ being twice as much, so the distance, $s = \bar{v}\Delta t$, increases 2x2=4 times).
The bottom line is, defining kinetic energy to be proportional to the square of the speed has turned out to make a useful theory.
Moreover, if the dependency was linear this would mean that the kinetic energy would vary by substituting $\mathbf{v}$ by $-\mathbf{v}$. When the second cannon fires the spring into the wall, traveling at twice the speed, the 24 cm spring compresses to a length of 4 cm (80% compressed); this of course requires 4 times more energy than the first collision with the wall. I will show how the energy of movement embodied in the object goes up from 0 to 1 to 4 to 9, etc. Now place another identical ball, m3, just to the right of m1 and also travelling at v=1 and with a compressed spring, s2, between them. Simply repeat the process to make m1 go from v=2 to v=3 by pushing off another identical ball, m4.
In other words, if I have two gas samples, both at the same temperature, then the average kinetic energy for the collection of gas molecules in one sample is equal to the average kinetic energy for the collection of gas molecules in the other sample. The amount of kinetic energy which an object has depends upon two variables: the mass (m) of the object and the speed (v) of the object. The focus here is Newton's first law of motion - sometimes referred to as the law of inertia. In addition to keeping the chassis off the track, race car suspension springs or roll-bars serve the same basic purposes as passenger car springs. When you are driving near the limit of grip you really want to spread the weight evenly otherwise overloading the front or rear tire can cause you to lose grip. Braking really upsets the cornering balance and will nearly always cause problems especially if hard braking is required. Expert drivers overlap the different phases of braking, turning and applying power on the tires.
Heel and toe shifting is an essential technique to master here as it allows you to brake and change down simultaneously while avoiding transmission shock loads which can unbalance the car and cause unwanted weight transfers. If you find yourself increasing or correcting the steering lock as you're traveling through the corner after the initial turn-in you've probably taken the wrong line. This isn't to say you can't retain a constant speed, but the important factor is that the car is in a neutral state until after the apex. It is also the tightest part of a corner and the point at which you are closest to the inside of the corner. The geometric apex of a constant radius corner is the central point of the corner, which may also be the racing apex, depending on the context.

I think it's the fastest way around the corner and when you get the car working for you as you want, the fastest maximum speed can be found that way. In picture above, the green line is a constant radius 90 degree right hander and the geometric apex is exactly half way around the corner. In addition to increasing the corner radius, this also involves taking a line which allows the earliest possible point of getting back into the throttle. It will make possible to apply brakes without strong steering input with smaller risk of unsettling the car stability. When approaching a tight apex slow up and aim to take the late apex of the bend leaving you a long gentle arc from this apex point. This approach works for corners which require hard accelerating cornering out of them, which will be most of them. The action of braking causes a weight transfer in the vehicle, shifting more weight from the rear of the car forward to the front tires, increasing the normal force on them and in turn increasing the amount of traction the front wheels have. However if you're in a lighter less powerful car you can apply the gas much closer to the apex point.
Because every track is different, it's important to analyze the racing lines through the first corner using video recordings from recent years. Of course, every driver tries to approach the first corner on precisely the ideal line, which as a rule is an extremely narrow corridor and only has space for one car. As a result, the car in front can now only change position once to prevent the car behind from overtaking. But the drivers like to integrate them into their driving line so they can make corner wider and be really fast. This allows you to lift objects and lower them with very little effort, so long as the mass-times-height added over all the objects is constant before and after.This is Archimedes' principle.
Therefore, it is natural to think that if you multiply both quantities, you need to end up with something like $K v^{2}$, where $K$ is an 'arbitrary' constant.
Therefore, the next simplest choice should be that the Lagrangian must contain the velocity squared. First, work out the total energy of the two ball and spring system before the spring is released. Unlike velocity, the kinetic energy of an object is completely described by magnitude alone. When you lift off the throttle or brake the weight is thrown forward, when you accelerate the weight is thrown to the rear. In understeer the front end breaks free first and the centrifugal force makes the car run wide.
By using all the space on the track, you can usually create a straighter route, and thus keep the speed up. Early apex maximizes the use of speed from the incoming straight; this is used when approaching a corner where the following straight is a lot shorter than the straight before the corner, or if we have to outbreak somebody during the overtake. To do this, the car must be straightening back out on the corner exit path as early as possible. By doing this you are turning a long sweeping bend into a tight bend with a long straight and this means that you will have more chance to accelerate. The point at which you can apply full power depends on your car and racing line you are using. However, there are many types of combinations of corners which require some analysis and preparation before race to understand the best approach. Because of the characteristics of weight transfer, this technique causes weight to be shifted away from the rear of the car, resulting in lower rear traction, and can be used to induce oversteer in some cases. So the drivers have to concentrate fully on the perfect position and the right braking point and also watch very carefully what is happening in front, behind and next to them. Unfortunately, to get access to all news, interviews and to open the site completely you should be subscribed to Autosport magazine. I do think that the Lagrangian is more fundamental in nature than the other quantities, however, its derivation involves the definition of work or equivalently, energy. The kinetic energy of both balls is the same and is KE=1 because all of the potential energy of the spring has been transferred symmetrically to the balls. There is no racing line and stuff like that, just adjusts your speed, don't brake during cornering and bear in mind weight transfer.
The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. They control the pitch and roll attitudes of the car and, in concert with the shock absorbers, the rate at which these attitudes change in response to the forces of lateral, longitudinal, and vertical acceleration.
Using a trailing (semi open) throttle you can keep the corner grip and take the corner at the fastest speed possible. They make effort to best utilize the traction circle, meaning they try to make the tire move as long as possible when applying pressure. However, there are various additional factors which determine the perfect racing line which are discussed below. Leave it too late and you'll miss the apex, too soon and you'll have to tighten your line mid corner. You can use trail braking or trailing (semi open) throttle to carry more speed to the corner, or use again heel and toe to ad to the balance. Once you have hit the apex you should be able to reduce the steering lock and increase the throttle. Blue line in the picture above is the late apex and this is generally regarded as the best strategy for racing, with a slightly lower entry speed but a faster exit speed.
Some cars will be able to apply full power straight after the apex, depending on the severity of the corner. In these few seconds at the absolute limit, one thing is clear: anyone who makes even the tiniest mistake can not only completely ruin his own chances, but also those of his team mates or his rivals.
So probably you won't buy the idea that this last explanation is the true cause of having the kinetic energy increasing quadratically, although, I think it is much more satisfactory than the first explanation. Finally, again with the shocks, they keep the tires in contact with the road surface as constantly as possible.
These techniques should only be applied on the track - it is foolish to push your car to the limits on the roads without the safety of run off areas. With skilful use of steering, brake and throttle (heel and toe), the driver can avoid spin. The entry and exit affects the speed into and out of the corner onto the following straight.
By now you should already be thinking about the next corner and position your car appropriately to allow you to use the racing line.
In no other phase of the race is the pressure so great, or must they absorb and process so much information at the same time.
Good techniques learnt for the track will one day come in handy when you hit a slippery road or if road conditions change suddenly. It's a sensible strategy to brake earlier as you're learning the track and your car, and progressively shorten the braking area as your experience grows.
Remember that the apex may be further round the corner than you can see, so make sure you learn the track and the apex points before driving in anger.
The shocks control the release of the energy stored in the springs when they are compressed.
The rule of thumb is to get most of the braking out of the way before turning into a corner, although a light brake pressure on entry can help to reduce understeer and can give a better turn in (this is known as trail braking). The amount of grip available is the factor which determines how late you can brake and apex. Racing imposes more stringent requirements than road drive, which include better accuracy of manufacture, minimum practical size, and weight, along with absolute consistency of performance.
And with use of heel and toe technique skilled driver can turn oversteer to understeer and vice versa. To realize the performance benefits offered by sophisticated shock absorbers, the springs must be absolutely consistent in load at height, rate, and linearity.