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A roller coaster is called a roller coaster because coasting is what it does, after it starts it continues coasting throughout the track. Many of the calculations rely on velocity in one way or another. Velocity is speed in a given direction. In order to know an objects velocity, you must know the speed and direction in which the object is traveling. To find velocity, you divide distance by time ( V=d/t ) The velocity is used to find acceleration. Which is the change in velocity divided by change in time.

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Kinetic means to move and energy means the ability to move Thus kinetic energy is the energy of motion. For example the faster the object moves the more kinetic energy is produced. The greater the speed and mass of an object the more kinetic energy there will be!

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Potential Energy is the same as stored energy. It is the energy of position. The greater the height the greater the potential energy. The stored energy is held within the gravitational field. For example, a lift motor, from a roller coaster, exerts kinetic energy when lifting the cart to the top of the hill as the cart moves up the hill the potential energy increases. At the top of the hill the cart has a huge amount of potential energy. As the cart is being dropped, the potential energy decreases, and kinetic energy is gained.

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Roller coasters operate because of conservation of energy ET = KE + PE Mechanical energy (ET) on a roller coaster comes in two forms. KE= Kinetic energy (1/2 mv 2 ) PE= Potential energy (mgh) To find the total energy you add the potential and kinetic energies together at the beginning of the ride. If we ignore friction than that is the total energy found at any location on the track.

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Solution:ET(top of 1 st hill)= ET(top of 2 nd hill) (KE + PE) hill 1 = (KE + PE) hill 2 [(1/2)mv 2 + mgh] hill 1 = [(1/2)mv 2 + mgh ] hill 2 the masses cancel because it is the same throughout the ride [(1/2)v 2 + gh] hill 1 = [(1/2)v 2 + gh] hill 2 substitute the numbers at each location (1/2)(.88) 2 + 9.8(.95) = (1/2)v 2 + 9.8(.65) notice all the numbers on the left side come from the top of the 1 st hill while all the numbers on the right side come form the top of the 2 nd hill.7744+ 9.31 = (1/2)v 2 + 6.37 3.7144 = (1/2)v 2 7.4288 = v 2 v= 2.73 M/s …. At the top of the 2 nd hill

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Now Lets Do Some Math!

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Height at the top of our coaster- 47cm=.47m Height at the lowest point of our coaster- 3.5cm=.035m Mass=55g Initial velocity=0 m/s (velocity at top) (KE + PE) top = (KE + PE) bottom [(1/2)mv 2 + mgh] top =[(1/2)mv 2 + mgh] bottom

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Velocity 1/2(55g)(0) 2 +(55g)(9.8m/s 2 )(.47m) = 1/2(55g)(v) 2 +(55g)(9.8m/s 2 )(.035m) …CALCULATING…

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Velocity= 2.92m/s

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Free-fall rides are really made up of three distinct parts: the ride to the top, the momentary suspension, and the downward plunge.

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Galileo first introduced the concept of free fall. His classic experiments led to the finding that all objects free fall at the same rate, regardless of their mass. According to legend, Galileo dropped balls of different mass from the Leaning Tower of Pisa to help support his ideas.

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The speed at which an object is falling during free fall can be determined, when started at rest, by this equation: V=g* t or velocity=acceleration of gravity x the change of time. The distance an object has traveled vertically during free fall can be determined, when started at rest, by this equation: D=1/2 g t 2 vertical distance= 1/2g (time) 2. Since horizontal distance is independent from vertical distance we can say the horizontal distance =vt

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We will assume the cart starts at the top of the hill at a velocity of 7 m/s The exciting hill is 100 meters tall Graphing the ordered pair as following (Horizontal distance, vertical distance)or (vt,.5gt 2 ) This means that for the 1 st second the cart will fall 4.9 meters. (7,4.9) At 2 seconds (14, 19.6); at 3 seconds (21,44.1) etc

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(Meters) This is the Free Fall of an object traveling at an initial speed of 7 meters per second!!!!!!!!!

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Notice the shape of the graph To increase the thrill of the ride, roller coaster designers often make the track follow the same shape as the graph. This makes the riders feel like they are free falling.

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Roller coasters like this one, on the left, use steep drops to build up tremendous speeds quickly, giving the rider the feeling of free fall.

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In the beginning of the ride, force is applied to the car to pull it to the top of the first big hill. The amount of force that must be applied depends on the mass of the car and its passengers. The force is applied by motors, and there is a built-in safety allowance. So the beginning is not the most thrilling…..

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The car is attached to this track, which gradually curves toward the ground. A stretch of straight track allows the car to slow down and brake, producing a controlled stop at the bottom, that keeps passengers from getting injured. So you cant have the most thrilling part of your roller coaster at the end…… So where does the most thrilling part of the ride occur……… The top of the first hill.

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As the car pulls it self to the highest point of the roller coaster, it suddenly drops and begins to accelerate toward the ground under the influence of the earth's gravity. The plunge seems dramatic. Just as Galileo and Newton explain in their theories of free fall, the least massive and most massive riders fall to the earth with the same rate of acceleration.

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The reason the most thrilling part is at the top of the roller coasters first hill is because that is the highest point you will be throughout the whole ride. Once you come racing down the big hill at the top you are going so fast that you dont realize you are only a few feet from the ground. Therefore, the most thrilling part of the roller coaster is right when you hit the top of the first hill.

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And …… dont forget to throw your hands up when you get to the top…………

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Loops.

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Centripetal Acceleration V²/r gives you centripetal acceleration.

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Circular vs. Clothoid Loops Clothoid LoopCircular Loops

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Notice as the roller coaster enters the loop it has a radius of 15 meters. At the top of the loop it has a radius of 5 meters. Then as the roller coaster exits the loop it has a radius of 20 meters. Since the centripetal acceleration is A=v 2 /r, the smaller the radius the more the acceleration. This makes it possible to make it around the loop at a smaller initial velocity. Clothoid loop 15 m 20m 5m

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The Ninja Weasel

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Which loops have greater thrills? Clothoid or circular loops? Clothoid loops are used on the Ninja Weasel because they require a smaller velocity than circular loops. At the top of the loop the radius is smaller this means the accelerations is greater than it would be for a circular loop.

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Warning: G s can Kill!!! Force of one (uno) of Earths atmospheric pressures. Amusement parks focus on getting roller coasters to produce the maxim amount of Gs while staying in the safe zone. Amusement parks focus on getting roller coasters to produce the maxim amount of Gs while staying in the safe zone. The safe zone for even high G rollercoaster is between 3.5 to 6.3 gs The safe zone for even high G rollercoaster is between 3.5 to 6.3 gs

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The highest amount of Gs ever survived by a human being was by John Stapp. The highest amount of Gs ever survived by a human being was by John Stapp. He survived a G force of 46.2 times the force of gravity for a little less than a second. He survived a G force of 46.2 times the force of gravity for a little less than a second. Remember exposure for more than a minute to 25+ Gs will leave life long bodily injury and even death. Remember exposure for more than a minute to 25+ Gs will leave life long bodily injury and even death. G-Facts

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Formula for finding Gs A/9.8m/s 2 Centripetal Acceleration divided by 9.8 meters per second squared For centripetal equations subtract one G from the top or add one to the bottom. Example equation:

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Example equation We have a random rollercoaster at the top of a traditional loop. We know gravity and are going to use a random acceleration of (55 m/s 2 ). Notice the gs felt change according to your position on the roller coaster(top or bottom of the loop).

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On the top to the loop you subtract 1 g from the number of gs calculated

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