Question

Suppose r(t)=costi+sintj+3tk represents the position of a particle on a helix, where z is the height of the particle above the ground.
a. What is t when the particle has height 16?
b. What is the velocity of the particle when its height is 16?
c. When the particle has height 16, it leaves the helix and moves along the tangent line at the constant velocity found in part b. Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line.

Answer 1

a. The $\vec k$ component tells you the particle's height:

$3t=16\implies t=\dfrac{16}3$

b. The particle's velocity is obtained by differentiating its position function:

$\vec v(t)=\dfrac{\mathrm d\vec r(t)}{\mathrm dt}=-\sin t\,\vec\imath+\cos t\,\vec\jmath+3\,\vec k$

so that its velocity at time $t=\frac{16}3$ is

$\vec v\left(\dfrac{16}3\right)=-\sin\dfrac{16}3\,\vec\imath+\cos\dfrac{16}3\,\vec\jmath+3\,\vec k$

c. The tangent to $\vec r(t)$ at $t=\frac{16}3$ is

$\vec T(t)=\vec r\left(\dfrac{16}3\right)+\vec v\left(\dfrac{16}3\right)t$