Question

(a) Consider a particle which starts moving around from the origin in a 3-dimensional space. De- termine the velocity vector v(t) in terms of φ and θ if it is constantly moving at the speed 5m/s, along the direction (φ,0). Here, φ denotes the angle between the z-axis and the projection of the position vector r(t) on the xy-plane; meanwhile θ denotes the angle between the z-axis and r(t). You may assume that (φ, θ) are fixed over time at this stage. Hence or otherwise write down the acceleration vector a(t). (b) Find the particle's position in terms of t when θ-φ-0°. and (φ,0)-(0°,90°). Repeat for (φ,0)-(900,00) (c) Suppose the angles are no longer fixed but in the form of time functions, such as, φ(t) and θ(t). If the particle is still moving at the speed as prescribed, derive the functions φ(t) and θ(t) in terms of a(t), y(t) and z(t) (d) Let r(t)-[3cos (nt), 3 sin(Tt), t]T. (i) Find φ(t) and θ(t). Hence, write down functions f1(φ),/2(d), fs(d) such that s(p) (ii) (Bonus*) Find g1 (φ, θ),92(φ.0), g3(φ, θ) such that (φ, θ) *If you solve the bonus question correctly, you will get an additional 5%. That is, at most 35% will be scored from Question 3. However, your assignment grade is still capped at 100%

Answer 1

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