Hi there i need help with these two questions before my exam next week, if anyone can help that would be great:
firstly this one:
A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10
− t)i + (t^2 − 10t)j + sin tk.
i. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.
ii. Find the curvature of the path, at t = 2π ≈ 6.28 seconds.
iii. Find the angle between the path (at start and end-points) and the k-direction.
And this one (obviously sketching might be tough)
A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.
i. Sketch the path of the particle.
ii. Find the value of t* at which the particle has greatest distance from (0, 0, 0).
iii. Show that at position r(t* ), the particles velocity is not perpendicular to r(t*).