A Mathematical Ride!

-Simran.M.Karkera.

That girl with the pink scarf would always visit the amusement park for one ride, the roller coaster. For she saw something beyond the thrill and adrenaline rush, she saw Mathematics in it! All her friends mocked her for her crazy love for mathematics. However, she silenced all her critics in the most classic way and showed them how the very foundation of the roller coaster is based on math. She embarked on a journey of proving her point and in the process explained to her peers what you are about to read below!

1.Designing: The designing of the roller coaster begins with something as basic as sketching graphs! Engineers make use of computer aided design(CAD) to ensure accurate maps of the hills, the twists and turns etc of the roller coaster.

a)Thrill: The thrill of a drop is the product of the angle of steepest descent in the drop (in radians) and the total vertical distance in the drop. The thrill of the coaster is the sum of the thrills of each drop.

Few constraints/conditions are formulated when designing the thrill. For e.g.

• The total horizontal length of the straight stretch must be less than 200 feet.
• The track must start 75 feet above the ground and end at ground level.
• At no time can the track be more than 75 feet above the ground or go below ground level.
• No ascent or descent can be steeper than 80 degrees from the horizontal.
• The roller coaster must start and end with a zero-degree incline.
• The path of the coaster must be modeled using differentiable functions.

These conditions or constraints will ensure two things:

1) Safety of the ride is maintained.

2) The forces like gravity, inertia etc. are properly balanced to ensure a smooth ride.

Now, in order to calculate the steepest descent, we use derivatives. We find the maxima (using the 2 conditions i.e. f(x)=0 and f’’(x)<0). Then, in order to determine the angle of steepest descent, we convert slope measurement into angle measurement. Finally, after multiplying the 2 values we come to thrill of one single drop! Continuing in similar fashion, the thrill of each of the drop is determined and added to arrive at the thrill of the roller coaster.

b) Loops:

The formula used to design a loop is:

a_c = v²⁄ r, Where,

• a_c is the acceleration,
• v is the velocity and
• r is the radius.

It is helpful to think of acceleration in terms of how many times the acceleration due to gravity is. So 1g is 10m=s2, 2g is 20m=s2, etc. At about 5-6g, especially upwards, a normal person will begin to pass out. So, the shape of the loops is to be chosen in such a way that the acceleration is not more than 4 or 5 mg. This becomes possible in a clothoid-shaped loop. The shape is a simple tear drop turned upside down with a radius more than that of the circular loops. This not just ensures less g-force but also helps in maintaining the sufficient speed required to keep the ride moving along the coaster.

2. The Vehicle:

The girl explained, “Although this may seem as something to be very easy to make and simple, it is rather complex as all the working related to the thrill, safety and the path would fail if the vehicle is not designed properly!

Here it’s not just math that is used here but also her cousin physics!”

“Wait, why do you call physics math’s cousin?”, asked the confused friends.

 “For the simple reason that mathematics is a very important tool used in physics while physics is a rich source of inspiration and insight in mathematics. For example: Before giving a mathematical proof for the formula for the volume of a sphere, Archimedes used physical reasoning to discover the solution (imagining the balancing of bodies on a scale).”

Coming back to the point, the vehicular math is also as essential as the path’s mathematics.

a) Speed: Firstly, we need to understand that along-with with the gravitational force, the coaster will have 2 types of energy acting on it: Kinetic and Potential. When the ride descends from the peak, it is gravity that pulls the vehicle down. Now coming to the other two aspects, kinetic energy will be the highest at the lowest point of the track while potential energy is the highest at the peak of the coaster. These facts can be converted into equations which is: At the point x in the picture, the potential energy lost is mgy, so the kinetic energy at that point is given by:

Solving for v gives us a basic formula:

To find a formula for t, we must manipulate the above equation. Rewriting v in terms of its horizontal and vertical components, let:

• s represent the accumulated distance along the curve,
• ds represent a small increment distance along the curve,
• dx and dy represent the horizontal and vertical components of ds. 

Substituting this into our basic formula:

Thus, t is:

We can use this formula on a roller coaster track with the equation:

Y=2cosx+sin2x

First we differentiate the equation:

To find time taken to travel from 0m to 1m:

Therefore the time taken to travel from 0m to 1m is about 0.24s

Finally, the speed will be the distance divided by time which in our example would be 1/0.24 m/second.

Mathematics is present everywhere around us right from the games that we play to the complex mechanisms and systems that operate in every country. You only have to take a deeper look at things and see how the magic of maths can turn things easier than you ever thought!

References:

https://www.intmath.com/help/snlabs/integr-proj4/integ-proj4.htm

659-Roller-Coaster-Math.pdf

https://www.teachengineering.org/lessons/view/duk_rollercoaster_music_less