Maths review: the arc length

Why do we have to learn math? How it is important in our daily life? What is calculus used for? All these questions a person can ask. Indeed, math is all around us. It is almost awkward to think of something that isn’t made without the benefit of mathematics. Where everything depend on numbers, sums, and measurements. Moreover, our life depend on algebra and geometry derived from Calculus. Luckily, architecture and calculus are linked together, where architecture do not give up on Calculus.

Architecture utilize calculus in many domains, especially in construction, where both Calculus and Architecture resort for geometry and algebra most of the time to evaluate the size, style, volume, and area of constructing.

For our project, we are highlighting on the usage of Math in knowing the distance and originating airplanes. Imagine we can know the direction of airplane, the length of altitude through using Math. Moreover, we can also find the appropriate angles between towers and planes by using certain equations.

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But how can a heavy airplane fly when anything thrown up move toward straight back down? In past time, they tried to invent tying wings to their arms or even putting wings on bicycle, until some people came and invented the first powered flight. With the support of math, they learnt about forces, weight, and lift that allows planes to fly. According to Newton’s Second Law, they also knew how to use thrust and knew what forces obliged to move the plane forward. Where all these helped in expand prospect in using Calculus.

Moreover, there are many secrets behind numbers where they helped us in discovering the miles and distance that takes the airplane from one country to another.

Shedding light on arc length is important in deriving it in order to notice the distance taken through planes. Using Math, there are now up to 12,000 aircrafts flying in the world on any day, proving again that math is really around us.

The Equation

We can take notice that the main idea of the equation depends on the integrals, and the derivatives. Both are foundational working tools in calculus that helped scientists and mathematicians discovering the world. For example the integrals can be useful and effective in calculating: the area between curves, the distance, velocity and acceleration, average value of a function, kinetic energy, surface area, and most importantly the arc length.

This is the main formula of the arc length and as we see, it is the combination between both integral and derivative. We have to take the borders of the arc first, from the start to the end, and then integrate the square root of the derivative of the function squared and then simply add it by 1.

The idea behind the arc length is dividing the arc in the graph into segments. The more we increase the number of the segments for the arc, the more we get a better and more accurate estimated length. But what will happen if the number of the segments approached infinity.

In order to get the distance for the airplane path we will have to imagine it in the two dimensional world, or in other words on paper. By looking at the airplane traveling from the side, it will form a curve. And every curve has an equation, but due to the fact that it is quite hard to get the right equation for such curve due to the plane movement and takeoff and landing etc… that’s why we managed to cut the curve into three parts.

First the takeoff part, which is the hardest. This part of the curve will have to be slightly increasing with a concave up looking curve and then just before it reaches the maximum height (10 km ≈) it will have to bend slightly to a constant value of y which is equal to the maximum value of height which is 10 km. This period will take ≈ 15 km to be completed. Now let’s assume that the landing part is just like the takeoff part, 15 km before it reaches the airport the landing process will start and it will have the inverted part of the takeoff, but this means that the arc length is going to be the same even if we flipped it upside down. Thus we will have to get the arc length of the takeoff equation and then multiply it by 2, while for the constant part we just have to get the distance.

Takeoff Landing Equation

This equation depends on exponentials due to its curve on graph. We just needed to play with the values for it to make the height and the distance covered for the takeoff part equal to 10 km. After all we ended up in this value: y=10e^(-x)-20e^(-x/2)+10

After derivation we got; y^'=10e^(-x/2)-10e^(-x) Now we just need to plot in the values in the arc length equation. ∫_0^10▒〖√(1+(10e^(-x/2)-10e^(-x) )^2 ) dx〗

By applying Simpson’s method: The value of this arc will be 11.66 km.

Assuming that the landing part will be equal, thus 23.5 km will be the value of both part. After all, the sum of the three parts will be equal to the arc length created by the airplane path from Beirut to New York, which is equal to 9003.5 km

All in all, as you have seen the arc length and calculus in specific have a significant role in almost all sciences even aviation. The arc length has been measured between to different countries located in two far away continents. that is almost very complicated and requires more mathematical techniques to measure the exact distance but what we can conclude an approximate number. 9003.5 is the distance covered from an airplane departing from Beirut airport to New York’s airport with the help of the arch length aviation controls and pilots can determine the time of arrival and the speed in which the airplane have travelled? Over the years many scientists have discovered numerous theories that can explain many physical phenomena's in the world. But the arc length is one the most important tool for a student, employee and scientist because it can be applied in various disciplines like accounting, architecture, aviation, physics and engineering to discover many secrets in this world. As have been illustrated in the second body paragraph as in the explanation of the arc length. The segments of the arc can be critical because the more we split the curve the more we reach an accurate result but as mentioned it's very difficult to receive the exact number due to the different and difficult techniques that not all students can do it. Calculus is a core subject in all universities around the world studying engineering because it approaches most engineering discipline with a logical and conceptual understanding of numbers.