Brain Gains (Rapid Recall, Jivin' Generation)

  1. A rover follows the path given by $\vec r(t) = (3t,t^2)$. Find the velocity of the rover at $t=2$.
  2. Give a vector equation of the tangent line to the rover's path at $t=2$, so $$\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}?\\?\end{pmatrix} t+\begin{pmatrix}?\\?\end{pmatrix}.$$
  3. Give the rover's speed at any time $t$.
  4. An object travels along straight lines. Its velocity is $ (0,3,4) $ m/s for 2 seconds, and then turns so its velocity is $ (1,2,-2) $ m/s for 5 seconds. Show the total distance (arc length) traveled by the object is 25m. [Hint: find the speed from the velocity.]

Group Problems

  1. Find the arc length of the curve $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$. Recall the arc length formula is $$\int_C ds = \int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$ Actually compute any integrals you encounter.
  2. Draw the vector field $\vec F(x,y) = \langle2x+y,x+2y\rangle$. (Based at $(x,y)$, draw the vector $\langle2x+y,x+2y\rangle$.)
  3. Draw the vector field $\vec F(x,y) = \langle y,-x\rangle$.
  4. Draw $\vec r(t) = (3 \cos t, 3 \sin t)$.
    • The curve above represents the position of an object. Compute the velocity of the object, so $\vec v(t) = \frac{d\vec r}{dt}$.
    • State the speed of the object above (simplify your answer to get a speed of 3). What is the difference between velocity and speed?
  5. Draw $\vec r(t) = (3 \cos 2t, 3 \sin 2t)$. (Suggestion - use multiples of $\pi/4$ for a table, rather than $\pi/2$. Why?) What is the speed of this curve? (Simplify the speed to get 6.)
  6. Draw $\vec r(t) = (7 \cos 5t, 7 \sin 5t)$. What is the speed of this curve? (Did you get 35?)
  7. Hurricane Matthew has a diameter of 28 miles. Assuming the eye is at the origin $(0,0)$, give a parametrization of the exterior edge of the hurricane.
    • Sustained winds are 128 mi/hr. Modify your parametrization above so that the speed is 128 mi/hr.
    • The eye of the hurricane is moving north west at a speed of 12 mi/hr. Modify your parametrization so that the center moves north west at 12 mi/hr.
  8. Let $\vec F=(-10,0)$ N and $\vec d=(2,1)$ m. Recall the projection of $\vec F$ onto $\vec d$ is $\ds \text{proj}_\vec d\vec F = \frac{\vec F\cdot \vec d}{\vec d\cdot \vec d}\vec d$.
    • Compute the projection of $\vec F$ onto $\vec d$ (so compute $\vec F_{\parallel \vec d}$).
    • Draw $\vec F$, $\vec d$ and $\text{proj}_\vec d\vec F $ on the same grid, all with their base at the origin. Try your best to give the $x$ and $y$ directions the same scale, otherwise you won't be able to see the connections among vectors.
    • Add to your picture the vector difference $\vec F_{\perp \vec d}=\vec F - \text{proj}_\vec d\vec F $. Which vectors in your picture are orthogonal?
    • Draw $\text{proj}_\vec F\vec d $, without doing any computations. Have each group member do this, and discuss any differences.