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When taking corners quickly, the biggest worry most drivers should have is slipping and losing control of the car. This happens when a driver takes the corner too fast. The physics of taking a flat corner depends on the equation vmax = Sqrt(mu*r*g). mu, the coefficient of static friction, is constant, as is g, the acceleration due to gravity. Therefore, a driver trying to take a corner as quickly as possible would like to make the radius of the turn as large as possible to allow for a higher vmax, keeping his car from slipping at higher speeds. But how? Doesn't a road have a defined radius? Yes, and no. The picture explains it. The arrow in the figure is what's called a "line" this is the best possible way for a car to take a corner at the highest speed. The line a regular driver would take is very curved, mimicking the road, and not allowing for a high vmax due to the small radius. A race car driver would take a better line. The racer's line is significantly less curved than the regular driver's line, making the radius much larger, allowing for a higher vmax . The racecar driver starts and ends wide of the inside and hits the apex of the turn, allowing for the least curved line possible. To conclude, when trying to take a corner quickly, the driver of the car should start out wide, hit the apex, and end wide, causing a relatively high radius and a relatively high vmax, without having the car slip off the road.
My friends and I go to Zumba classes three times a week and it is very fun. Like any regular physics student, I am constantly thinking about Mr. Fullerton's lessons during class. As we dance, jump, and move I get to thinking... it must take a lot of energy to move around the way we do. But as we eat healthily and exercise more often, Zumba gets easier and easier... why? Here are some of the equations I will be using to help explain this Zumba Paradox... - KE = (1/2)(mass)(velocity2) - PE = (mass)(g)(height) - Work = Change in Mechanical Energy - Work = Force * Displacement It takes work to move our body in all different sorts of ways. Because work is equal to the change in Mechanical Energy, and both Kinetic Energy and Potential Energy are proportional to the mass of the object, it is reasonable to say that work is also proportional to the mass of the object. In this case, the object is our body. As any athletic trainer will happily tell you, a good workout is one where you do the most work. In our case, we will hold everything else constant besides our mass because we are doing the exact same class every time we work out. Put extremely simply, work is how much you move times how much weight you are moving. So, it is correct to say that as you lose mass you will do less and less work each successive time you go to Zumba class. I want to lose weight at a constant rate, as would most females in Zumba class. Constant weight loss is much better than fluctuating weight loss. So how can I keep my weight loss constant, and overcome this work-mass relationship that we discussed earlier? Zumba deals with changes in Kinetic Energy more than other types of fitness training such as weight lifting which deals more with changes in Potential Energy. So for simplicity we will set Work equal to the change in only KE. Here's what we want to happen: C = (1/2)(mass)(velocity2) // With C being a constant positive number that represents an amount of Joules In order for us to keep a constant C, velocity2 has to increase at a rate equal to the rate at which mass decreases. Here's our relationship in equation form: velocity2 = 1/mass // or in exponential form --> velocity = mass-1/2 So there it is, ladies and gents, if you want to lose weight at a constant rate, you need to increase your intensity a little bit each class as you shed the pounds.