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SIT204: Mathematical Model of the Motion of the Fire Pot - Mathematics Assignment
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Internal Code: MAS3543
You must provide a derivation of your model - not simply a statement of the final equations - using the mathematics covered in this unit. Your derivation should contain all relevant mathematical working, and any necessary explanation of methods or algorithms as appropriate. It is suggested that you implement the following procedure to develop your model:
1) Create a coordinate system aligned to the plane of rotation.
a. Define the coordinates of the center of rotation, C, such that is between 1 and 1.5 meters above the ground (and located anywhere in the XY plane).
b. Define a unit normal vector to the plane of rotation, n^.
Questions 2:
a) Obtain an equation for the trajectory of the firepot, from the moment it is released until it would strike the ground, in the world frame of reference. This will require you to determine the initial conditions for the trajectory (position and velocity in the world frame) using your model developed in Part 1, and then use these to solve for the constants of integration
when developing the trajectory equation.
b) Using your trajectory model, determine the following quantities as functions of the initial conditions (given as variable quantities):
1. maximum height reached by the firepot
2. maximum range of the fire pot from the point of release
c) Determine the rotation speed and time t of release, such that your model from Part 1 would generate an initial position and velocity that would have the firepot acheive a height of 3 meters at a distance of at least 20 meters from the launch point.
Question 3:
The ground within the goblin scene is known as a terrain mesh: a set of polygons formed by points (called vertices) that are regularly spaced in their XY coordinates in the ground plane of the world frame, as in the image below (for example).
In this problem you are to define a small set of vertices as a section of a terrain mesh. These vertices should be regularly spaced in the XY plane, with varying heights. Create a set of triangles from these vertices and use your points to demonstrate the following:
a) Given values for the X and Y coordinates of a randomly chosen location within the boundary of the ground plane under your mesh (not a vertex location of the mesh), determine a method to find the triangle covering this point. Determine the height of the
mesh at this location.
b) Choose two points on the ground plane under your mesh and define the line (in the XY plane) between these two points. Segment this line into a set of lines that lie under each of the triangles of the mesh above the line (blue segments below). For each line segment, determine the equation of a line in 3D space that lies in the mesh (green segments) and for which its projection into the ground plane is the given blue line segment.
c) Using your line segments that lie across the mesh, create a plot of height on the line segments above the ground plane, as a function of distance along the line. You should end up with a continuous curve that is piecewise linear.
Mathematics Assignment Help
Questions 1: You are to create a mathematical model of the motion of the fire pot as it is swung in a circle by the goblin. Your model must describe the trajectory of the fire pot in the camera frame of reference: that is, provide and equation for the displacement of the fire pot, as a function of time t, the initial conditions of the motion and the rotation speed, ? (in revolutions per second), relative to the origin of the camera frame. Consider the diagram below, which shows quantities relevant to the mathematical description of the problem.
You must provide a derivation of your model - not simply a statement of the final equations - using the mathematics covered in this unit. Your derivation should contain all relevant mathematical working, and any necessary explanation of methods or algorithms as appropriate. It is suggested that you implement the following procedure to develop your model:
1) Create a coordinate system aligned to the plane of rotation.
a. Define the coordinates of the center of rotation, C, such that is between 1 and 1.5 meters above the ground (and located anywhere in the XY plane).
b. Define a unit normal vector to the plane of rotation, n^.
Questions 2:
a) Obtain an equation for the trajectory of the firepot, from the moment it is released until it would strike the ground, in the world frame of reference. This will require you to determine the initial conditions for the trajectory (position and velocity in the world frame) using your model developed in Part 1, and then use these to solve for the constants of integration
when developing the trajectory equation.
b) Using your trajectory model, determine the following quantities as functions of the initial conditions (given as variable quantities):
1. maximum height reached by the firepot
2. maximum range of the fire pot from the point of release
c) Determine the rotation speed and time t of release, such that your model from Part 1 would generate an initial position and velocity that would have the firepot acheive a height of 3 meters at a distance of at least 20 meters from the launch point.
Question 3:
The ground within the goblin scene is known as a terrain mesh: a set of polygons formed by points (called vertices) that are regularly spaced in their XY coordinates in the ground plane of the world frame, as in the image below (for example).
In this problem you are to define a small set of vertices as a section of a terrain mesh. These vertices should be regularly spaced in the XY plane, with varying heights. Create a set of triangles from these vertices and use your points to demonstrate the following:
a) Given values for the X and Y coordinates of a randomly chosen location within the boundary of the ground plane under your mesh (not a vertex location of the mesh), determine a method to find the triangle covering this point. Determine the height of the
mesh at this location.
b) Choose two points on the ground plane under your mesh and define the line (in the XY plane) between these two points. Segment this line into a set of lines that lie under each of the triangles of the mesh above the line (blue segments below). For each line segment, determine the equation of a line in 3D space that lies in the mesh (green segments) and for which its projection into the ground plane is the given blue line segment.
c) Using your line segments that lie across the mesh, create a plot of height on the line segments above the ground plane, as a function of distance along the line. You should end up with a continuous curve that is piecewise linear.