Homework 3

Due: February 6, 2018, 8 a.m.

Please give a complete, justified solution to each question below. A single-term answer without explanation will receive no credit.

Please complete each question on its own sheet of paper (or more if necessary), and upload to Gradsescope.

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\dydx}{\frac{dy}{dx}} \newcommand{\proj}{\textrm{proj}} % For boldface vectors: \renewcommand{\vec}[1]{\mathbf{#1}} $$

1.  

  • Find the point on the curve $$\vec r(t) = \langle t^3 + 3t, t^2 + 1, \ln(1 + 2t) \rangle, ~ 0 \leq t \leq \pi,$$ where the tangent line is orthogonal to the plane $$15 x + 4 y + 0.4 z = 10.$$
  • Then find the intersection of this tangent line with the plane above.

2. A spaceship, with its engines engaged, has position vector $$ \vec{r}(t) = \left\langle t \cos (t),t \sin (t),a t\right\rangle$$ where $a$ can be set as an arbitrary constant. \] The space station is located at coordinates $\langle -\pi-3,-3\pi,2\rangle$. The captain wants to coast into the space station.

  • When should the engines be turned off? (Hint: You can use onely the $x$- and $y$-components to do this)
  • What value should $a$ be set to?

3. Show that a projectile fired from ground attains three-quarters of its maximum height in half the time it takes to reach the maximum height.

4. Suppose an ant crawls onto the edge of a 12\" (diameter) record on a turntable spinning at $33\frac13$ revolutions per minute. A stationary observer looking from above will see its path as motion in the $xy$-plane.

Ant on a record

  • Suppose the ant stops on the edge of the spinning platter (radius $6''$). Find its position, velocity and acceleration relative to the observer as vector-valued finctions of time $t$ in seconds. (You can model its starting position as $\langle 6,0\rangle$ at time $t=0$.)
  • Now imagine the ant walks down a radial line of the record toward the center of the turntable. From its perspective, it walks at constant speed and in a straight line, arriving at the center at time $t=45$. Write the ant's position as a vector-valued function of time $t$.
  • At any given time $t$, let $\vec b$ be the unit vector pointing from the ant's position toward the center of the turntable, and let $\vec c$ be the unit vector in the direction of the spinning turntable (i.e., perpendicular to $\vec b$). Write the acceleration of the ant as a linear combination of $\vec b$ and $\vec c$.