Need help with Lines, Planes, Curves and Trajectories, Vectors ,algebra homework help

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Would like if possible the best formula to work out the problems and step by step how you solved them.

Q1 and Q2 are related to the Intro file.

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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, w (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. nxù û (t) fo d 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, în. C. Determine a vector û that lies in the plane of rotation, that is both orthogonal to în and parallel to the ground plane of the world. d. Obtain a third unit vector ûl, that is orthogonal to both î and û, such that you have a right-handed orthonormal basis {n,û, û } aligned to the plane. 2) Determine a counter-clockwise parameterisation of a circle of radius r in your plane. a. Determine a parametric equation (in parameter t) describing the angular position 8 of point P (the position of the firepot) relative to the reference line û. This equation should involve the initial angular displacement at t = 0 (i.e., 6.) and the angular speed w, given in revolutions per second. b. Using the basis vectors {ù,ūl}, define the equation of the circle as a vector-valued function in the plane (which should involve angle 6). C. Substitute your parametric equation for 8 (t) to obtain coordinates of P in the plane, as a function of initial conditions r, 60, w and time t. d. Use your coordinates in the plane to state a vector parametric equation for r(t), in the basis {n, û û1} 3) Obtain a vector-valued equation for the displacement of the firepot, in the camera frame of reference a. Define either a set of equations, or a transformation matrix, to convert vectors from the basis {n, û, û } to the external basis {i, j, k} of the world frame b. Define either a set of equations, or a transformation matrix, to convert vectors from the basis {i, j, k} to the basis {f, r, d}, being the forward, right and down orthonormal basis vectors of the camera frame. C. Define the position of the camera in the world frame and use this to determine a vector c from the position of the camera to the center of rotation, C. d. Convert both vectors c and r(t) into the camera frame and add them together, to obtain the equation for the displacement of the point P from the origin of the camera frame. Question 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 firepot 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.
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Explanation & Answer

I added the solution for those parameters. In the process I found a mistake (previously I chose the vector u in opposite direction to the desired).

0. The terms.
̂ . Also
Denote the plane of rotation 𝑅, the center of rotation 𝐶, the unit normal vector to 𝑅 at 𝐶 as 𝒏
̂ and parallel to the 𝑥𝑦-plane as 𝒖
̂ . The angle between the
denote the unit vector perpendicular to 𝒏
radius-vector 𝒓 will be called 𝜃 and the angle corresponding to time 𝑡 = 0 will be 𝜃0 . Let the radius
of rotation be 𝑟, the (constant) angular speed be 𝜔.
The position of 𝑅 relative to the world frame of reference.
̂ makes with the 𝑥𝑦-plane and the angle 𝜓 that the
It may be specified by the angle 𝜑 that 𝒏
̂ to the 𝑥𝑦-plane makes with the 𝑥 (to the right) axis.
projection of 𝒏

a. Denote the coordinates of the point 𝐶 as (0, 0, ℎ) in the standard world frame of reference. It is
given that 1 ≤ ℎ ≤ 1.5 (in meters).

̂ on the 𝑧-axis is sin 𝜑, the projection on the 𝑥𝑦-plane has the length cos 𝜑
b. The projection of 𝒏
hence the 𝑥-projection is cos 𝜑 cos 𝜓 and the 𝑦-projection is cos 𝜑 sin 𝜓.
̂ = cos 𝜑 cos 𝜓 𝒊 + cos 𝜑 sin 𝜓 𝒋 + sin 𝜑 𝒌.

̂ lies in 𝑅 (parallel to it). If 𝑅 is not horizontal (which I believe is
c. Any vector that is orthogonal to 𝒏
true), then there is only one unit vector in it which is parallel to the ground (𝑥𝑦-plane). Denote it 𝒖
̂ = 𝑎𝒊 + 𝑏𝒋 and 𝒖
̂ , i.e. 𝑎 cos 𝜑 cos 𝜓 + 𝑏 cos 𝜑 sin 𝜓 = 0 or
We know that 𝒖
̂ = − sin 𝜓 𝒊 + cos 𝜓 𝒋.
𝑎 cos 𝜓 + 𝑏 sin 𝜓 = 0. Because we need a unit vector, 𝒖

̂ and 𝒖
̂ , use cross product 𝒖
̂⊥ = 𝒏
d. To obtain a third vector orthogonal to both 𝒏
It lies in 𝑅 because it is orthogonal to 𝒏
̂⊥ = 𝒏
̂ = |cos 𝜑 cos 𝜓
− sin 𝜓

cos 𝜑 sin 𝜓
cos 𝜓

sin 𝜑| = − sin 𝜑 cos 𝜓 𝒊 − sin 𝜑 sin 𝜓 𝒋 + cos 𝜑 𝒌.

̂ and 𝒖
̂ ⊥ make an orthonormal basis in 𝑅.
This way the vectors 𝒖

̂ , as 𝜃0 . The
a. Denote the angle corresponding to 𝑡 = 0, which the rope makes with the vector 𝒖
angular speed 𝜔 in radians per second is supposed to be constant.
Then the angular displacement is linear: 𝜃(𝑡) = 𝜃0 + 𝜔𝑡.

̂, 𝒖
̂ ⊥ of a vector 𝒗
̂ in 𝑅 that makes angle 𝜃 with the vector 𝒖
̂ are obviously
b. The projections on 𝒖
𝑟 cos 𝜃 and 𝑟 sin 𝜃. This means
̂ = 𝑟 cos 𝜃 ∙ 𝒖
̂ + 𝑟 sin 𝜃 ∙ 𝒖

c. In terms of time 𝑡 we obtain
̂ + 𝑟 sin(𝜃0 + 𝜔𝑡) ∙ 𝒖
𝑃(𝑡) = 𝑟 cos(𝜃0 + 𝜔𝑡) ∙ 𝒖

d. Almost the same:
̂ + 𝑟 cos(𝜃0 + 𝜔𝑡) ∙ 𝒖
̂ + 𝑟 sin(𝜃0 + 𝜔𝑡) ∙ 𝒖
𝒓( 𝑡 ) = 0 ∙ 𝒏
Or we can write it as
𝒓(𝑡) = (𝑟 cos(𝜃0 + 𝜔𝑡))
𝑟 sin(𝜃0 + 𝜔𝑡)
̂, 𝒖
̂, 𝒖
̂ ⊥ }.
with respect to the basis {𝒏

̂, 𝒖
̂, 𝒖
̂ ⊥ } in terms of {𝒊, 𝒋, 𝒌}. The required matrix consists of these
a. We know how to express {𝒏
decompositions as columns:
cos 𝜑 cos 𝜓
𝐴 = ( cos 𝜑 sin 𝜓
sin 𝜑

− sin 𝜓
cos 𝜓

− sin 𝜑 cos 𝜓
− sin 𝜑 sin 𝜓 ).
cos 𝜑

b. The set of equations is obviously 𝒊 = 𝒓̂, 𝒋 = 𝒇̂, 𝒌 = −𝒅
0 1
Hence the transformation matrix is 𝐵 = (1 0
0 0

0 ).

c. Well, let the camera be 𝑚 meters behind the point of rotation: (0, −𝑚, ℎ) in world frame.
The vector to 𝐶 is (0, 𝑚, 0).

d. To convert 𝒓(𝑡) known in 𝑅 coordinates we need to perform the transformation given by 𝐴 and
then by 𝐵:
− sin 𝜓 𝑟 cos(𝜃0 + 𝜔𝑡) − sin 𝜑 cos 𝜓 𝑟 sin(𝜃0 + 𝜔𝑡)
(𝑟 cos(𝜃0 + 𝜔𝑡)) ↦ ( cos 𝜓 𝑟 cos(𝜃0 + 𝜔𝑡) − sin 𝜑 sin 𝜓 �...

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