91Ó°ÊÓ

The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path \(y=\frac{1}{3} x^{3}(x\) and \(y\) are measured in miles) can safely go 30 miles per hour at \(\left(1, \frac{1}{3}\right)\). How fast can it go at \(\left(\frac{3}{2}, \frac{9}{8}\right)\) ?

Let \(C\) be a curve given by \(y=f(x)\). Let \(K\) be the curvature \((K \neq 0)\) at the point \(P\left(x_{0}, y_{0}\right)\) and let \(z=\frac{1+f^{\prime}\left(x_{0}\right)^{2}}{f^{\prime \prime}\left(x_{0}\right)}\) Show that the coordinates \((\alpha, \beta)\) of the center of curvature at \(P\) are \((\alpha, \beta)=\left(x_{0}-f^{\prime}\left(x_{0}\right) z, y_{0}+z\right)\).

Air Traffic Control Because of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 4 miles to make a \(90^{\circ}\) turn and climb to an altitude of \(4.2\) miles. The model for the path of the plane during this maneuver is \(\mathbf{r}(t)=\langle 10 \cos 10 \pi t, 10 \sin 10 \pi t, 4+4 t\rangle, \quad 0 \leq t \leq \frac{1}{20}\) where \(t\) is the time in hours and \(\mathbf{r}\) is the distance in miles. (a) Determine the speed of the plane. (b) Use a computer algebra system to calculate \(a_{\mathbf{T}}\) and \(a_{\mathrm{N}}\) Why is one of these equal to 0 ?

Centripetal Force An object of mass \(m\) moves at a constant speed \(v\) in a circular path of radius \(r .\) The force required to produce the centripetal component of acceleration is called the centripetal force and is given by \(F=m v^{2} / r .\) Newton's Law of Universal Gravitation is given by \(F=G M m / d^{2}\), where \(d\) is the distance between the centers of the two bodies of masses \(M\) and \(m\), and \(G\) is a gravitational constant. Use this law to show that the speed required for circular motion is \(v=\sqrt{G M / r}\)

State the definition of continuity of a vector-valued function. Give an example of a vector-valued function that is defined but not continuous at \(t=2\).

A 5500 -pound vehicle is driven at a speed of 30 miles per hour on a circular interchange of radius 100 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires?