1. ## calc 3, motion

Consider the motion of a particle along a helix given by r(t) = sinti + costj + (t^2 - 3t + 2)k, where the k component measures the height in meters above ground and t >= 0.
(a) Does the particle ever move downward?
(b) Does the particle ever stop moving?
(c) At what times does it reach a position 12 meters above the ground?
(d) What is the velocity of the particle when it is 12 meters above the ground?

2. a) For this question you're only interested in the k component as that measures the up-and-down-ness.

If the particle is moving downwards, it means its vertical velocity (rate of change of the k component w.r.t. t) is negative.

So, differentiate the k component w.r.t. t (that is, the $\displaystyle t^2 - 3t + 2$ and see whether what you get can ever be negative.

b) For it to stop moving, the components of i, j and k must all have zero velocity.

So differentiate all components w.r.t. t and see if there's a value of t that will make all of these zero at the same time. My suggestion is that I rather think there isn't.

c) Simply solve the equation in t that puts the k component equal to 12. You'll have a quadratic to solve.

d) Plug the value of t into tha equation you got in a) and see what you get.