91Ó°ÊÓ

The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=(6-t) \mathbf{i}+t \mathbf{j},(3,3) $$

Consider the vector-valued function \(\mathbf{r}(t)=t \mathbf{i}+\left(4-t^{2}\right) \mathbf{j}\) (a) Sketch the graph of \(\mathbf{r}(t)\). Use a graphing utility to verify your graph. (b) Sketch the vectors \(\mathbf{r}(1), \mathbf{r}(1.25)\), and \(\mathbf{r}(1.25)-\mathbf{r}(1)\) on the graph in part (a). (c) Compare the vector \(\mathbf{r}^{\prime}(1)\) with the vector \(\frac{\mathbf{r}(1.25)-\mathbf{r}(1)}{1.25-1}\).

A baseball is hit 3 feet above the ground at 100 feet per second and at an angle of \(45^{\circ}\) with respect to the ground. Find (a) the vector-valued function for the path of the baseball, (b) the maximum height, (c) the range, and (d) the arc length of the trajectory.

The graph of the vector-valued function \(\mathbf{r}(t)\) and a tangent vector to the graph at \(t=t_{0}\) are given. (a) Find a set of parametric equations for the tangent line to the graph at \(t=t_{0}\) (b) Use the equations for the tangent line to approximate \(\mathbf{r}\left(t_{0}+\mathbf{0 . 1}\right)\) $$ \mathbf{r}(t)=\left\langle t, \sqrt{25-t^{2}}, \sqrt{25-t^{2}}\right\rangle, \quad t_{0}=3 $$

Investigation Consider the helix represented by the vectorvalued function \(\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t, t\rangle\) (a) Write the length of the arc \(s\) on the helix as a function of \(t\) by evaluating the integral $$ s=\int_{0}^{t} \sqrt{\left[x^{\prime}(u)\right]^{2}+\left[y^{\prime}(u)\right]^{2}+\left[z^{\prime}(u)\right]^{2}} d u . $$ (b) Solve for \(t\) in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter \(s\). (c) Find the coordinates of the point on the helix for arc lengths $$ s=\sqrt{5} \text { and } s=4 $$ (d) Verify that \(\left\|\mathbf{r}^{\prime}(s)\right\|=1\).

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} \\ \mathbf{v}(1)=5 \mathbf{j}, \quad \mathbf{r}(1)=\mathbf{0} \end{array} $$

Use the model for projectile motion, assuming there is no air resistance. 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.

Use the model for projectile motion, assuming there is no air resistance. The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 4 feet. The pass is released at an angle of \(35^{\circ}\) with the horizontal. (a) Find the speed of the football when it is released. (b) Find the maximum height of the football. (c) Find the time the receiver has to reach the proper position after the quarterback releases the football.

Use the model for projectile motion, assuming there is no air resistance. A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector. (a) Find the minimum initial speed of the bale and the corresponding angle at which it must be ejected from the baler. (b) The ejector has a fixed angle of \(45^{\circ} .\) Find the initial speed required for a bale to reach its target.

A particle moves on a straight-line path that passes through the points (2,3,0) and \((0,8,8) .\) Find a vector-valued function for the path. Use a computer algebra system to graph your function. (There are many correct answers.)