Calculating Speed With Roller Cars Part A: Fill & Download for Free

iding in a CarWhen you are riding in a car and the car accelerates, your body tends to move backward against the seat. Likewise, if the car stops suddenly, your body tends to move forward, in the direction of the dashboard. Note the language here: "tends to move" rather than "is pushed." To say that something is pushed would imply that a force has been applied, yet what is at work here is not a force, but inertia—the tendency of an object in motion to remain in motion, and an object at rest to remain at rest.A car that is not moving is, by definition, at rest, and so is the rider. Once the car begins moving, thus experiencing a change in velocity, the rider's body still tends to remain in the fixed position. Hence, it is not a force that has pushed the rider backward against the seat; rather, force has pushed the car forward, and the seat moves up to meet the rider's back. When stopping, once again, there is a sudden change in velocity from a certain value down to zero. The rider, meanwhile, is continuing to move forward due to inertia, and thus, his or her body has a tendency to keep moving in the direction of the now-stationary dashboard.This may seem a bit too simple to anyone who has studied inertia, but because the human mind has such a strong inclination to perceive inertia as a force in itself, it needs to be clarified in the most basic terms. This habit is similar to the experience you have when sitting in a vehicle that is standing still, while another vehicle alongside moves backward. In the first split-second of awareness, your mind tends to interpret the backward motion of the other car as forward motion on the part of the car in which you are sitting—even though your own car is standing still.Now we will consider the effects of centripetal force, as well as the illusion of centrifugal force. When a car turns to the left, it is undergoing a form of rotation, describing a 90°-angle or one-quarter of a circle. Once again, your body experiences inertia, since it was in motion along with the car at the beginning of the turn, and thus you tend to move forward. The car, at the same time, has largely overcome its own inertia and moved into the leftward turn. Thus the car door itself is moving to the left. As the door meets the right side of your body, you have the sensation of being pushed outward against the door, but in fact what has happened is that the door has moved inward.The illusion of centrifugal force is so deeply ingrained in the popular imagination that it warrants further discussion below. But while on the subject of riding in an automobile, we need to examine another illustration of centripetal force. It should be noted in this context that for a car to make a turn at all, there must be friction between the tires and the road. Friction is the force that resists motion when the surface of one object comes into contact with the surface of another; yet ironically, while opposing motion, friction also makes relative motion possible.Suppose, then, that a driver applies the brakes while making a turn. This now adds a force tangential, or at a right angle, to the centripetal force. If this force is greater than the centripetal force—that is, if the car is moving too fast—the vehicle will slide forward rather than making the turn. The results, as anyone who has ever been in this situation will attest, can be disastrous.The above highlights the significance of the centripetal force requirement: without a sufficient degree of centripetal force, an object simply cannot turn. Curves are usually banked to maximize centripetal force, meaning that the roadway tilts inward in the direction of the curve. This banking causes a change in velocity, and hence, in acceleration, resulting in an additional quantity known as reaction force, which provides the vehicle with the centripetal force necessary for making the turn.The formula for calculating the angle at which a curve should be banked takes into account the car's speed and the angle of the curve, but does not include the mass of the vehicle itself. As a result, highway departments post signs stating the speed at which vehicles should make the turn, but these signs do not need to include specific statements regarding the weight of given models.The CentrifugeTo return to the subject of "centrifugal force"—which, as noted earlier, is really just centrifugal motion—you might ask, "If there is no such thing as centrifugal force, how does a centrifuge work?" Used widely in medicine and a variety of sciences, a centrifuge is a device that separates particles within a liquid. One application, for instance, is to separate red blood cells from plasma.Typically a centrifuge consists of a base; a rotating tube perpendicular to the base; and two vials attached by movable centrifuge arms to the rotating tube. The movable arms are hinged at the top of the rotating tube, and thus can move upward at an angle approaching 90° to the tube. When the tube begins to spin, centripetal force pulls the material in the vials toward the center.Materials that are denser have greater inertia, and thus are less responsive to centripetal force. Hence, they seem to be pushed outward, but in fact what has happened is that the less dense material has been pulled inward. This leads to the separation of components, for instance, with plasma on the top and red blood cells on the bottom. Again, the plasma is not as dense, and thus is more easily pulled toward the center of rotation, whereas the red blood cells respond less, and consequently remain on the bottom.The centrifuge was invented in 1883 by Carl de Laval (1845-1913), a Swedish engineer, who used it to separate cream from milk. During the 1920s, the chemist Theodor Svedberg (1884-1971), who was also Swedish, improved on Laval's work to create the ultracentrifuge, used for separating very small particles of similar weight.In a typical ultracentrifuge, the vials are no larger than 0.2 in (0.6 cm) in diameter, and these may rotate at speeds of up to 230,000 revolutions per minute. Most centrifuges in use by industry rotate in a range between 1,000 and 15,000 revolutions per minute, but others with scientific applications rotate at a much higher rate, and can produce a force more than 25,000 times as great as that of gravity.In 1994, researchers at the University of Colorado created a sort of super-centrifuge for simulating stresses applied to dams and other large structures. The instrument has just one centrifuge arm, measuring 19.69 ft (6 m), attached to which is a swinging basket containing a scale model of the structure to be tested. If the model is 1/50 the size of the actual structure, then the centrifuge is set to create a centripetal force 50 times that of gravity.The Colorado centrifuge has also been used to test the effects of explosions on buildings. Because the combination of forces—centripetal, gravity, and that of the explosion itself—is so great, it takes a very small quantity of explosive to measure the effects of a blast on a model of the building.Industrial uses of the centrifuge include that for which Laval invented it—separation of cream from milk—as well as the separation of impurities from other substances. Water can be removed from oil or jet fuel with a centrifuge, and likewise, waste-management agencies use it to separate solid materials from waste water prior to purifying the water itself.Closer to home, a washing machine on spin cycle is a type of centrifuge. As the wet clothes spin, the water in them tends to move outward, separating from the clothes themselves. An even simpler, more down-to-earth centrifuge can be created by tying a fairly heavy weight to a rope and swinging it above one's head: once again, the weight behaves as though it were pushed outward, though in fact, it is only responding to inertia.Roller Coasters and Centripetal ForcePeople ride roller coasters, of course, for the thrill they experience, but that thrill has more to do with centripetal force than with speed. By the late twentieth century, roller coasters capable of speeds above 90 MPH (144 km/h) began to appear in amusement parks around America; but prior to that time, the actual speeds of a roller coaster were not particularly impressive. Seldom, if ever, did they exceed that of a car moving down the highway. On the other hand, the acceleration and centripetal force generated on a roller coaster are high, conveying a sense ofweightlessness (and sometimes the opposite of weightlessness) that is memorable indeed.Few parts of a roller coaster ride are straight and flat—usually just those segments that mark the end of one ride and the beginning of another. The rest of the track is generally composed of dips and hills, banked turns, and in some cases, clothoid loops. The latter refers to a geometric shape known as a clothoid, rather like a teardrop upside-down.Because of its shape, the clothoid has a much smaller radius at the top than at the bottom—a key factor in the operation of the roller coaster ride through these loops. In days past, roller-coaster designers used perfectly circular loops, which allowed cars to enter them at speeds that were too high, built too much force and resulted in injuries for riders. Eventually, engineers recognized the clothoid as a means of providing a safe, fun ride.As you move into the clothoid loop, then up, then over, and down, you are constantly changing position. Speed, too, is changing. On the way up the loop, the roller coaster slows due to a decrease in kinetic energy, or the energy that an object possesses by virtue of its movement. At the top of the loop, the roller coaster has gained a great deal of potential energy, or the energy an object possesses by virtue of its position, and its kinetic energy is at zero. But once it starts going down the other side, kinetic energy—and with it speed—increases rapidly once again.Throughout the ride, you experience two forces, gravity, or weight, and the force (due to motion) of the roller coaster itself, known as normal force. Like kinetic and potential energy—which rise and fall correspondingly with dips and hills—normal force and gravitational force are locked in a sort of "competition" throughout the roller-coaster rider. For the coaster to have its proper effect, normal force must exceed that of gravity in most places.The increase in normal force on a roller-coaster ride can be attributed to acceleration and centripetal motion, which cause you to experience something other than gravity. Hence, at the top of a loop, you feel lighter than normal, and at the bottom,

A useful near-answer is provided by the bicycle motorpacing speed record, currently held by Fred Rompleberg at 269 km/h: http://www.fredrompelberg.com/en/html/algemeen/fredrompelberg/record.aspIn that record, the rider is drafting behind a vehicle so tightly that wind resistance is negligible, and the most important countering force is rolling resistance.It's an imperfect comparison: on one hand there is still air around the bike, and that may be creating non-trivial wind resistance (not least on the tops of the wheels, which travel 2x the ground speed), and there could possibly be some vacuum effect partly drawing the rider towards the pace car (though Rompelberg was still pedaling flat-out; at best any vacuum effect was just reducing the effects of rolling resistance).In terms of rolling resistance, here's a starter on some plausible numbers and formulae:http://en.wikipedia.org/wiki/Rolling_resistance...but I decline to turn that data into a "real-world" calculation of when rolling resistance would hit the limits of human performance (which you can figure on being around 600W for excellent cyclists over durations in this range; that's what Sam Wittingham, HPV record holder, claims for his sustained wattage, which was good for 130 km/h in the flying 200m (that's the human-powered vehicle land speed record).Update: It appears Bruce Bursford set a roller-riding record of 334 km/h:http://www.bikebrothers.co.uk/ultimatebike.htmhttp://en.wikipedia.org/wiki/Cycling_recordsNote that the first link, right at the bottom of the page, has a small photo showing what I believe is the roller setup. This is a sort of stationary bike rig, but in this case it involves a real bicycle with the wheels running on a set of rollers (avid cyclists will note that unlike most home-training rollers, Bursford's setup locked the bike upright on the rollers).This may get closer to a theoretical in-a-vacuum result, though of course he was still pedaling the fast-moving wheels through still air, surely reducing his ultimate wheel speed. I would say Bursford's performance establishes a minimum theoretical speed for riding in a vacuum, assuming you could cope with, um, all the other issues...why did I try to answer this question?

A roller coaster is started by towing the car up the initial incline to the first peak, which stores the energy required to propel the car to the end of the course. From then on, energy is traded back and forth between potential and kinetic energy as the car ascends and descends inclines. I will assume that the car is of negligible length and that its mass is constant.Energy is a scalar quantity, so the two quantities of interest are the speed of the car (the magnitude of the velocity) at any given position and its height. This is transformed into kinetic energy as the car accelerates down. For an ideal lossless roller coaster, the relationship between the height and speed at any two points along the course after the first peak is given by the equation:mg(h1) + ½m(v1)^2 = mg(h2) + ½m(v2)^2where: h1 = height at location 1v1 = speed at location 1h2 = height at location 2v2 = speed at location 2g = acceleration of gravitym = mass of carIf the speed at the summit is zero and we use the first trough as a height reference and assign its height as zero, them the potential energy at the top and the speed at the bottom are related by:mg(h1) = ½m(v2)^2The mass is present on both sides of equation so the relationship between height and speed can be simplified:g(h1) = ½ (v2)^2so the speed in the first trough is proportional to the square root of the height at the first peak.If these were the only factors affecting energy then the car could keep on going up and down forever, with no loss of height at the peaks. However, in the real world, energy losses occur, and these can be accounted for by adding an additional term to the equation:mgh1 + ½mv1^2 = mgh2 + ½mv2^2 + E where E is energy lostBecause of these energy losses, on each subsequent portion of the course the car is unable to get quite as high as it did on the preceding peak and its maximum speed in the troughs also decreases.Most of the energy is likely lost to two things:friction, which increases approximately proportional to the force perpendicular to the track,andaerodynamic drag, which increases as v^2.Other minor losses would be caused by such things as the deforming of the structure of the roller coaster under load.Frictional losses increase along with the load, so that losses will be highest when g forces are their highest, at the troughs in the ride, in accordance with the law of friction (which is not really a physical law in the usual sense but a rule of thumb which is an approximation good enough to be useful under certain circumstances). They also increase with increasing mass. Aerodynamic drag is related to the projected surface area of the object along with its shape, and increases as the square of the speed, so these losses are also at their highest in the troughs. Drag losses are not primarily related to the mass of the car.It’s important that the coaster designer understand how much energy will be lost in any segment of the course. Otherwise, he or she may encounter problems such as the following:If losses are higher than anticipated, the car could get stuck on the track because not enough energy remains to get the car up to the top of the next peak; andIf losses are less than anticipated, the speed over the peak could be greater than desired, with excessive negative g forces; and speed could be higher than anticipated in the trough, with too much positive g forces at the bottom, which could result in excessive forces on the passengers as well as on the structure.The designer is likely to be able to estimate energy losses closely on the basis of past experiences (his and his predecessors) building similar structures. However, the energy losses over any part of the structure can be determined from measurements of speed and height at various positions along the course. By measuring the height and speed of the car at any two points along the track and substituting the measured values into the equation:E = mg(h1 - h2) + ½m[(v1)^2 - (v2)^2]one can determine the value of E, the energy lost along that segment of the course. Additional measurements and calculations would allow determination of how much energy is being lost to air resistance, friction and other factors.Drag force is directly proportional to the mass density of the air. Atmospheric pressure drops by approximately 3% from the sea level value per 1000 feet of altitude up to about 10,000 feet and, other factors being equal, density is proportional to pressure. So if a roller coaster designed for operation at sea level were simply moved to a higher elevation then the aerodynamic drag would decrease and speeds would increase along the track.As mentioned above, the designer of a coaster would primarily be interested in controlling the energy in the car, either potential or kinetic, along the course, and so knowing the amount of energy loss, including that of aerodynamic drag, would be critical. In the case of an unmodified design transplanted to a significantly higher elevation, then speeds at some points might exceed some of the initial design parameters. I expect that the designer would modify the coaster design to take into account the decrease in energy loss, so that the modified design would have speeds along the course similar to that of the original design.The designer could approach this problem by obtaining a coefficient of drag for the car. Once this figure is available, then the aerodynamic drag force for a given mass density can be calculated and the coaster design modified to take into account the expected energy loss at any altitude.