# Math Help - particle question

1. ## particle question

Not sure if this is in the right forum

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.
Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.
Find the curvature of the path, at t = 2π ≈ 6.28 seconds.
Find the angle between the path (at start and end-points) and the k-direction.

And this one too, i think i have to optimise the square but i don't think i have done it right

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.

Find the value of t∗ at which the particle has greatest distance from (0, 0, 0).

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

Thanks heaps if you can help i can't get the right answers

2. I'm not sure about the first question but for the second you have to maximise
$|r(t)|^2$ (the square of the distance from the particle to the origin)

in other words find t such that $\frac{d |r(t)|^2}{dt}=0$ and $\frac{d^2|r(t)|^2}{dt^2}<0$

NB $|r(t)|^2=(2t+1)^2+(4-t^2)^2$

for the second part find r(t) at the time you found above then find the velocity of the particle v(t) at the time you found above

NB $v(t)=\frac{d}{dt}(r(t))$

Remember if two vectors are perpendicular then there dot product=0 so using those two vectors find it and show it doesn't equal 0