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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

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).

Hint: optimising squared distance may be the simpler method here.

iii. Show that at position r(t ∗ ), the particles velocity is not perpendicular to r(t∗ ).

From a practice exam, for my final this week