How Is Symmetry Used In Designing A Roller Coaster
Designing a roller-coaster is not easy, they have to be thrilling and exciting but not jerky. A lot of work has to be done to design a safe yet breathtaking roller coaster. The engineers have to consider the forces acting on the roller coaster train as they gain momentum and the structural support needed to support these forces. There are different designs of a roller coaster, for example, a suspended roller coaster, in which the track is above the speeding car. There are wooden, regular steel, steel-flying and many more types of roller coasters.
Designing a roller coaster is a blend of mathematics and physics, the two subjects that interests me a lot. Thus, the topic caught my attention.
For my math exploration, I would investigate the math …show more content…
behind the parabolicly designed roller coasters. Parabolicly designed roller coasters are interesting and most popular because when the coaster falls from the peak of the track, all the bodies are falling at the same rate, i.e. the track has the same gradient. The only force acting here is gravity.
To create a parabolic roller coaster, you should analyze the properties of the design from a mathematical perspective, based on the height, width and area. The fundamental properties of the designed parabola consist of the vertex, the point of symmetry and foci. The engineers place these points, by considering, the force of gravity and the velocity, acting upon particular points on the track.
A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
The general equation of a regular parabolic curve: y=ax^2+bx+c In the above equation, a, b and c are co-efficient where a is not equal to 0.
There are two types of parabolas: vertical and horizontal. For my exploration, I would be focussing on the vertical form of parabolas because through that, I would be able to investigate the effect of the gradient (slope) and the gravitational forces on the speed of the coaster.
The following image shows different aspects of a vertical parabola:
PS: In the image, a is greater than 0.
Vertex:
The points where a graph turns is called a vertex. Vertex indicates the maximum or minimum points of the parabolic curve. For the general equation of a parabola, the vertex can be calculated by the following formula: x=- b/2a
The equation for the “vertex” form of a parabola, with its vertex at (h, k), is: y=a〖(x-h)〗^2+k In most roller coasters, the vertex is a maximum point, which means the curve is concave up and the value of a, the co-efficient of x2, is positive and greater than 0.
The vertex, mainly the y coordinates of it, indicate the maximum or minimum height of the roller coaster.
Axis of symmetry:
The vertical line that passes through the vertex is called the axis of symmetry. Every parabola is symmetrical about its axis of symmetry. Axis of symmetry are the x value of the vertex. Axis of symmetry can be obtained through the following formula: x=- b/2a
Symmetry is a very important mathematical concept. Symmetry is vital in the construction or designing of different applications including roller coasters. As Dobre and Daniel state in their paper on symmetry, engineers design machines according to the principle “shape follows function”. An aspect of beauty is symmetry, which represents relative simplicity within complexity. Even simplistic things, such as logos, start with a basic rectangular or square shape and then the graphic artist use symmetry to create the design. The same applies to a design of a roller coaster. Symmetry also plays an important role in human visual perception and aesthetics. Symmetry is important for the safety aspect of a roller …show more content…
coaster.
Foci:
To form a parabola according to ancient Greek definitions, you would start with a line and a point off to one side.
The line is called the "directrix"; the point is called the "focus". The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus.
The conical form of a regular parabolic equation is as follows:
4p(y – k) =〖(x-h)〗^2
The equation is just another way of expressing the equation y= a(x-h)2+k, where (h,k) are the (x,y) coordinates of the vertex. P represents the difference between the vertex and the focus and the difference between the vertex and the directrix.
To design a roller coaster, foci is taken into consideration to measure the interior width of the parabola. Depending on the shape of the roller coaster (concave up or concave down), the focus changes as the cart travels up and down. The greater the foci, the steeper the parabola will be and faster the cart would go.
Designing a parabolic roller coaster (autograph):
Let’s suppose the maximum height of the roller coaster is 300m. The vertex is (100, 300). The axis of symmetry will be 300m as well. We can assume that a is
-1.
Thus, the equation will be : y=-〖(x-100)〗^2+300 The graph shows that the x-intercepts aGradient is the degree of inclination. The gradient of a roller coaster track is constantly changing due to the design of the track. It can be said, the greater the gradient (steepness), the greater the speed of the roller coaster, if other factors like air resistance are constant.
To calculate the gradient of a roller coaster, we need to start off by defining the track, with an equation. Let’s assume, the equation is y= 5x + 70. In order for the curve to be continuous and differentiable, x=0.
Since we are working with parabolas, the equation shall be y=ax^2+bx+c. f(0)= 70 c= 70 y=ax2+bx+70 f’(x)= 2ax + b In order to be differentiable, f’(x)= 2ax + b, must equal the slope of 5x+70, at x=0. 2ax+b= 5 When x=0, b=5. Let a=-1 y=-x2+5x+70
The above image shows the graph that we obtain from the equation that we generated, after differentiation.
We can calculate the gradient, at different points by the formula: m=(y^2-y^1)/(x^2-x^1 )
Let’s consider the coordinates (-5.97,4.43), (-4.97,20.5), (7.11,55.0) and (8.11,44.7)
The gradient for the first set of coordinates (-5.97,4.43), (-4.97,20.5) is 1.47.
The gradient for the second set of coordinates (7.11,55.0), (8.11,44.7) is -10.3.
The gradient on the other left side (descending) of the curve is negative that means the rate of change i.e. the acceleration decreases. The velocity, although increases, the rate at which it change is less.
However, in a roller coaster, gradient is one of the factors that affect the velocity of the cart. Factors, such as, the height of the track, air resistance and the gravitational force are equally important. Research shows that, when the roller coaster is falling, gravitational force is the most dominant factor that influences the velocity of the cart. Thus, although the acceleration is less, the passengers feel that it increases as the cart goes downward.
Height of the roller coaster:
It is important for engineers to draft the approximate height of the roller coasters before the construction begins. To determine the position vector of the roller coaster, a well known formula for the projectile motion is used.
Let’s assume, the following question has been presented:
Determine the position vector and find the maximum height of a launch roller coaster traveling with an initial velocity 60 mph of with a height of 15ft from the ground. The roller coaster is launched from an angle of 30◦ . r(t)=[V_0 cos〖(θ)t]〗 i+h+[V_0 sin(θ)t-1/2 gt^2]j V0 is the initial velocity h is the starting height θ is the angle above the horizontal at which the coaster is launched g is the acceleration due to gravity
The values given in the question will be subsituted in the equation: r(t)=[76.21t]i+[30+44t-16t^2]j Differentiate the equation to obtain velocity function:
r^'(t) =vt=[76.21]i+[44-32t]j
The roller coaster will be at its maximum point when the velocity is equal to 0. 44-32t=0 t = 1.375 seconds
By substituting the value of t into the previous equation, the answer is 60.25 ft (maximum height of the launcher)
Examining the different designs of a roller coaster:
As stated earlier, roller coasters have different designs and structures. Parabolas are one of them. Different concepts of mathematics can be used to construct the basic structure of a roller coaster, for example, sine curve, circular eclipse, etc.
Sine Curve:
The equation used to design a roller coaster on the basis of a roller coaster is as follows: y=A sin〖(bx-c)〗+d a is the amplitude of the sine curve b is the period of the sine curve c is the phase shift of the sine curve d is the vertical shift of the sine curve
Let’s assume, the amplitude of the track is 25m, the period is 2π. The phase shift and the vertical shift shall be 0.
The generated equation is:
y=25 sin〖(x)〗
Circle Ellipse:
The formulae used to design a circular shaped roller coaster include the circumference and area of the circle forumula. The circumference of the track is important to determine to number of carts/length of the carts that can move along the track at the same time.
The area and circumference can be calculated by simple formulas: The engineers and technicians can further work on the design to calculate the energy required to power the roller coaster and all the forces acting upon the track and the carts, for example, Centripetal acceleration that can be calculated by the following formula:
Conclusion:
I started off to research about the maths behind the designs of roller coasters. It is fascinating to learn how knowledge of different subjects blends itself to form one product. Physics, Mathematics and Arts play a significant part in the designing and construction of a roller coaster. When you go to the theme parks and ride in the roller coasters, all you can think about is the extent of fun that you have or how thrilling and scary the adventure is. We tend to overlook the fact that hundreds of people have worked hard and put in their sweat and blood to design the wonderful ride for our amusement. Through this exploration, I discovered the importance of every single, minute detail that goes behind the designing of a roller coaster to make it jerky yet safe for the passengers. I have not greatly explored the concept and importance of the forces that act upon the track and the carts of the roller coaster because it was related more to physics than maths, however, it is always interesting to learn about the effect of gravity on our surroundings. re 82.7 and 117.
Designing a roller coaster is a blend of mathematics and physics, the two subjects that interests me a lot. Thus, the topic caught my attention.
For my math exploration, I would investigate the math …show more content…
behind the parabolicly designed roller coasters. Parabolicly designed roller coasters are interesting and most popular because when the coaster falls from the peak of the track, all the bodies are falling at the same rate, i.e. the track has the same gradient. The only force acting here is gravity.
To create a parabolic roller coaster, you should analyze the properties of the design from a mathematical perspective, based on the height, width and area. The fundamental properties of the designed parabola consist of the vertex, the point of symmetry and foci. The engineers place these points, by considering, the force of gravity and the velocity, acting upon particular points on the track.
A parabola is defined as follows: For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.
The general equation of a regular parabolic curve: y=ax^2+bx+c In the above equation, a, b and c are co-efficient where a is not equal to 0.
There are two types of parabolas: vertical and horizontal. For my exploration, I would be focussing on the vertical form of parabolas because through that, I would be able to investigate the effect of the gradient (slope) and the gravitational forces on the speed of the coaster.
The following image shows different aspects of a vertical parabola:
PS: In the image, a is greater than 0.
Vertex:
The points where a graph turns is called a vertex. Vertex indicates the maximum or minimum points of the parabolic curve. For the general equation of a parabola, the vertex can be calculated by the following formula: x=- b/2a
The equation for the “vertex” form of a parabola, with its vertex at (h, k), is: y=a〖(x-h)〗^2+k In most roller coasters, the vertex is a maximum point, which means the curve is concave up and the value of a, the co-efficient of x2, is positive and greater than 0.
The vertex, mainly the y coordinates of it, indicate the maximum or minimum height of the roller coaster.
Axis of symmetry:
The vertical line that passes through the vertex is called the axis of symmetry. Every parabola is symmetrical about its axis of symmetry. Axis of symmetry are the x value of the vertex. Axis of symmetry can be obtained through the following formula: x=- b/2a
Symmetry is a very important mathematical concept. Symmetry is vital in the construction or designing of different applications including roller coasters. As Dobre and Daniel state in their paper on symmetry, engineers design machines according to the principle “shape follows function”. An aspect of beauty is symmetry, which represents relative simplicity within complexity. Even simplistic things, such as logos, start with a basic rectangular or square shape and then the graphic artist use symmetry to create the design. The same applies to a design of a roller coaster. Symmetry also plays an important role in human visual perception and aesthetics. Symmetry is important for the safety aspect of a roller …show more content…
coaster.
Foci:
To form a parabola according to ancient Greek definitions, you would start with a line and a point off to one side.
The line is called the "directrix"; the point is called the "focus". The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus.
The conical form of a regular parabolic equation is as follows:
4p(y – k) =〖(x-h)〗^2
The equation is just another way of expressing the equation y= a(x-h)2+k, where (h,k) are the (x,y) coordinates of the vertex. P represents the difference between the vertex and the focus and the difference between the vertex and the directrix.
To design a roller coaster, foci is taken into consideration to measure the interior width of the parabola. Depending on the shape of the roller coaster (concave up or concave down), the focus changes as the cart travels up and down. The greater the foci, the steeper the parabola will be and faster the cart would go.
Designing a parabolic roller coaster (autograph):
Let’s suppose the maximum height of the roller coaster is 300m. The vertex is (100, 300). The axis of symmetry will be 300m as well. We can assume that a is
-1.
Thus, the equation will be : y=-〖(x-100)〗^2+300 The graph shows that the x-intercepts aGradient is the degree of inclination. The gradient of a roller coaster track is constantly changing due to the design of the track. It can be said, the greater the gradient (steepness), the greater the speed of the roller coaster, if other factors like air resistance are constant.
To calculate the gradient of a roller coaster, we need to start off by defining the track, with an equation. Let’s assume, the equation is y= 5x + 70. In order for the curve to be continuous and differentiable, x=0.
Since we are working with parabolas, the equation shall be y=ax^2+bx+c. f(0)= 70 c= 70 y=ax2+bx+70 f’(x)= 2ax + b In order to be differentiable, f’(x)= 2ax + b, must equal the slope of 5x+70, at x=0. 2ax+b= 5 When x=0, b=5. Let a=-1 y=-x2+5x+70
The above image shows the graph that we obtain from the equation that we generated, after differentiation.
We can calculate the gradient, at different points by the formula: m=(y^2-y^1)/(x^2-x^1 )
Let’s consider the coordinates (-5.97,4.43), (-4.97,20.5), (7.11,55.0) and (8.11,44.7)
The gradient for the first set of coordinates (-5.97,4.43), (-4.97,20.5) is 1.47.
The gradient for the second set of coordinates (7.11,55.0), (8.11,44.7) is -10.3.
The gradient on the other left side (descending) of the curve is negative that means the rate of change i.e. the acceleration decreases. The velocity, although increases, the rate at which it change is less.
However, in a roller coaster, gradient is one of the factors that affect the velocity of the cart. Factors, such as, the height of the track, air resistance and the gravitational force are equally important. Research shows that, when the roller coaster is falling, gravitational force is the most dominant factor that influences the velocity of the cart. Thus, although the acceleration is less, the passengers feel that it increases as the cart goes downward.
Height of the roller coaster:
It is important for engineers to draft the approximate height of the roller coasters before the construction begins. To determine the position vector of the roller coaster, a well known formula for the projectile motion is used.
Let’s assume, the following question has been presented:
Determine the position vector and find the maximum height of a launch roller coaster traveling with an initial velocity 60 mph of with a height of 15ft from the ground. The roller coaster is launched from an angle of 30◦ . r(t)=[V_0 cos〖(θ)t]〗 i+h+[V_0 sin(θ)t-1/2 gt^2]j V0 is the initial velocity h is the starting height θ is the angle above the horizontal at which the coaster is launched g is the acceleration due to gravity
The values given in the question will be subsituted in the equation: r(t)=[76.21t]i+[30+44t-16t^2]j Differentiate the equation to obtain velocity function:
r^'(t) =vt=[76.21]i+[44-32t]j
The roller coaster will be at its maximum point when the velocity is equal to 0. 44-32t=0 t = 1.375 seconds
By substituting the value of t into the previous equation, the answer is 60.25 ft (maximum height of the launcher)
Examining the different designs of a roller coaster:
As stated earlier, roller coasters have different designs and structures. Parabolas are one of them. Different concepts of mathematics can be used to construct the basic structure of a roller coaster, for example, sine curve, circular eclipse, etc.
Sine Curve:
The equation used to design a roller coaster on the basis of a roller coaster is as follows: y=A sin〖(bx-c)〗+d a is the amplitude of the sine curve b is the period of the sine curve c is the phase shift of the sine curve d is the vertical shift of the sine curve
Let’s assume, the amplitude of the track is 25m, the period is 2π. The phase shift and the vertical shift shall be 0.
The generated equation is:
y=25 sin〖(x)〗
Circle Ellipse:
The formulae used to design a circular shaped roller coaster include the circumference and area of the circle forumula. The circumference of the track is important to determine to number of carts/length of the carts that can move along the track at the same time.
The area and circumference can be calculated by simple formulas: The engineers and technicians can further work on the design to calculate the energy required to power the roller coaster and all the forces acting upon the track and the carts, for example, Centripetal acceleration that can be calculated by the following formula:
Conclusion:
I started off to research about the maths behind the designs of roller coasters. It is fascinating to learn how knowledge of different subjects blends itself to form one product. Physics, Mathematics and Arts play a significant part in the designing and construction of a roller coaster. When you go to the theme parks and ride in the roller coasters, all you can think about is the extent of fun that you have or how thrilling and scary the adventure is. We tend to overlook the fact that hundreds of people have worked hard and put in their sweat and blood to design the wonderful ride for our amusement. Through this exploration, I discovered the importance of every single, minute detail that goes behind the designing of a roller coaster to make it jerky yet safe for the passengers. I have not greatly explored the concept and importance of the forces that act upon the track and the carts of the roller coaster because it was related more to physics than maths, however, it is always interesting to learn about the effect of gravity on our surroundings. re 82.7 and 117.