- I-Learn, Class Pictures, Learning Targets, Text Book Practice
- Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus
Prep
We're in Unit 1 - Motion. Your homework assignment each day is to spend 1-2 hours working on the next 4 tasks from the current unit's prep.
Brain Gains (Rapid Recall, Jivin' Generation)
- A rover follows the path given by $\vec r(t) = (3t,t^2)$. Find the velocity of the rover at $t=2$.
Solution
The velocity is the derivative, so we have $\vec r'(t) = (3,2t)$. At $t=2$, this is $\vec r'(2)=(3,4)$.
r[t_] := {3 t, t^2}
r'[t]
r'[2]
- Give a vector equation of the tangent line to the rover's path at $t=2$, so $$\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}?\\?\end{pmatrix} t+\begin{pmatrix}?\\?\end{pmatrix}.$$
Solution
The rover's position at $t=2$ is $\vec r(2)=(6,4)$. The velocity (direction vector for the line) is $\vec r'(2) = (3,4)$. An equation of the tangent line is is thus $$\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}3\\4\end{pmatrix} t+\begin{pmatrix}6\\4\end{pmatrix}.$$
We can plot the curve and tangent line together in Mathematica.
r[t_] := {3 t, t^2}
r2[t_] := r'[2] t + r[2]
r2[t]
Show[
ParametricPlot[r[t], {t, 1, 3}],
ParametricPlot[r2[t], {t, -1, 1}, PlotStyle -> Orange]
]
- Give the rover's speed at any time $t$.
Solution
The speed is the magnitude of the velocity, so $|\vec r'(t)| = \sqrt{3^2+(2t)^2}$.
The ComplexExpand[] command makes the assumption that all variables in the problem are real, which allows Mathematica to do extra simplification.
r[t_] := {3 t, t^2}
Norm[r'[t]]
Norm[r'[t]] // ComplexExpand
- 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.]
Solution
Distance is the product of speed and time. For the first two seconds, the distance traveled is $|(0,3,4)|2 =(5)(2)=10$. For the next 5 seconds, the distance traveled is $|(1,2,-2)|5 = (3)(5)=15$. The total distance traveled is then 10+15, which we get by summing these two distances.
Group Problems
- 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?
- 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.)
- Draw $\vec r(t) = (7 \cos 5t, 7 \sin 5t)$. What is the speed of this curve? (Did you get 35?)
- 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.
- 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.
- 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.
- Work on Task 9.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |