91Ó°ÊÓ

Find the vector-valued function for the path of a projectile launched at a height of 10 feet above the ground with an initial velocity of 88 feet per second and at an angle of \(30^{\circ}\) above the horizontal. Use a graphing utility to graph the path of the projectile.

Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of \(45^{\circ}\) above the horizontal.

A baseball, hit 3 feet above the ground, leaves the bat at an angle of \(45^{\circ}\) and is caught by an outfielder 3 feet above the ground and 300 feet from home plate. What is the initial speed of the ball, and how high does it rise?

Eliminate the parameter \(t\) from the position function for the motion of a projectile to show that the rectangular equation is \(y=-\frac{16 \sec ^{2} \theta}{v_{0}^{2}} x^{2}+(\tan \theta) x+h\)

Find the principal unit normal vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=\cos t \mathbf{i}+2 \sin t \mathbf{j}+\mathbf{k}, \quad t=-\frac{\pi}{4} $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\)

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\sqrt{t} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+\frac{1}{4} t \mathbf{k} $$

Determine the maximum height and range of a projectile fired at a height of \(1.5\) meters above the ground with an initial velocity of 100 meters per second and at an angle of \(30^{\circ}\) above the horizontal.

Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=3 x-2, \quad x=a $$

Consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=b(\omega t-\sin \omega t) \mathrm{i}+b(1-\cos \omega t) \mathbf{j}\) where \(\omega\) is the constant angular velocity of the circle and \(b\) is the radius of the circle. Find the maximum speed of a point on the circumference of an automobile tire of radius 1 foot when the automobile is traveling at 55 miles per hour. Compare this speed with the speed of the automobile.