Mathematical Immaturity

14.19 Exercises

1. A particle moves in a plane so that its position at time $t$ has polar coordinates $r = t,$ $\theta = t.$ Find formulas for the velocity $\mathbf{v},$ the acceleration $\mathbf{a},$ and the curvature $\kappa$ at any time $t.$
2. A particle moves in space so that its position at time $t$ has cylindrical coordinates $r = t,$ $\theta = t,$ $z = t.$ It traces out a curve called a conical helix. (a) Find formulas for the velocity $\mathbf{v},$ the acceleration $\mathbf{a},$ and the curvature $\kappa$ at time $t.$ (b) Find a formula for determining the angle between the velocity vector and the generator of the cone at each point of the curve.
3. A particle moves in space so that its position at time $t$ has cylindrical coordinates $r = \sin t,$ $\theta = t,$ $z = \log \sec t,$ where $0 \leq t < \frac{1}{2}\pi.$ (a) Show that the curve lies on the cylinder with Cartesian equation $x^2 + (y - \frac{1}{2})^2 = \frac{1}{4}.$ (b) Find a formula (in terms of $t$) for the angle which the velocity vector makes with $\,\mathbf{k}.$
4. If a curve is given by a polar equation $r = f(\theta),$ where $a \leq \theta \leq b \leq a + 2\pi,$ prove that the arc length is $$ \begin{align*} \\ \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta \end{align*} $$
5. The curve described by the polar equation $r = a(1 + \cos \theta),$ where $a > 0$ and $0 \leq \theta \leq 2\pi,$ is called a cardioid. Draw a graph of the cardioid $r = 4(1 + \cos \theta)$ and compute its arc length.
6. A particle moves along a plane curve whose polar equation is $r = e^{c\theta},$ where $c$ is a constant and $\theta$ varies from 0 to $2\pi.$ (a) Make a sketch indicating the general shape of the curve for each of the following values of $c:$ $c = 0,$ $c = 1,$ $c = -1.$ (b) Let $L(c)$ denote the arc length of the curve and let $a(c)$ denote the area of the region swept out by the position vector as $\theta$ varies from 0 to $2\pi.$ Compute $L(c)$ and $a(c)$ in terms of $c.$
7. Sketch the curve whose polar equation is $r = \sin^2 \theta,$ $0 \leq \theta \leq 2\pi,$ and show that it consists of two loops. (a) Find the area of region enclosed by one loop of the curve. (b) Compute the length of one loop of the curve.
8. In each of Exercises 8 through 11, make a sketch of the plane curve having the given polar equation and compute its arc length. $$ \begin{align*} r = \theta, \quad 0 \leq \theta \leq \pi. \end{align*} $$
9. In each of Exercises 8 through 11, make a sketch of the plane curve having the given polar equation and compute its arc length. $$r = e^\theta,\quad 0 \leq \theta \leq \pi.$$
10. In each of Exercises 8 through 11, make a sketch of the plane curve having the given polar equation and compute its arc length. $$r = 1 + \cos \theta,\quad 0 \leq \theta \leq \pi.$$
11. In each of Exercises 8 through 11, make a sketch of the plane curve having the given polar equation and compute its arc length. $$r = 1 - \cos \theta,\quad 0 \leq \theta \leq 2\pi.$$
12. If a curve has the polar equation $r = f(\theta),$ show that its radius of curvature $\rho$ is given by the formula \begin{align*} \\ \rho = \frac{(r^2 + r'^2)^{3/2}}{|r^2 + 2r'^2 - rr''|}, \end{align*} where $r' = f'(\theta)$ and $r'' = f''(\theta).$
13. For each of the curves in Exercises 8 through 11, compute the radius of curvature for the value of $\theta$ indicated. (a) Arbitrary $\theta$ in Exercise 8. (b) Arbitrary $\theta$ in Exercise 9. (c) $\theta = \frac{1}{4}\pi$ in Exercise 10. (d) $\theta = \frac{1}{4}\pi$ in Exercise 11.
14. Let $\phi$ denote the angle, $0 \leq \phi \leq \pi,$ between the position vector and the velocity vector of a curve. If the curve is expressed in polar coordinates, prove that $v \sin \phi = r$ and $v \cos \phi = dr/d\theta,$ where $v$ is the speed.
15. A missile is designed to move directly toward its target. Due to mechanical failure, its direction in actual flight makes a fixed angle $\alpha \neq 0$ with the line from the missile to the target. Find the path if it is fired at a fixed target. Discuss how the path varies with $\alpha.$ Does the missile ever reach the target? (Assume the motion takes place in a plane.)
16. Due to a mechanical failure, a ground crew has lost control of a missile recently fired. It is known that the missile will proceed at a constant speed on a straight course of unknown direction. When the missile is 4 miles away, it is sighted for an instant and lost again. Immediately an anti-missile missile is fired with a constant speed three times that of the first missile. What should be the course of the second missile in order for it to overtake the first one? (Assume both missiles move in the same plane.)
17. Prove that if a homogeneous first-order differential equation of the form $y' = f(x, y)$ is rewritten in polar coordinates, it reduces to a separable equation. Use this method to solve $y' = (y - x)/(y + x).$
18. A particle (moving in space) has velocity vector $\mathbf{v} = \omega\,\mathbf{k} \times \mathbf{r},$ where $\omega$ is a positive constant and $\mathbf{r}$ is the position vector. Prove that the particle moves along a circle with constant angular speed $\omega.$ (The angular speed is defined to be $|d\theta/dt|,$ where $\theta$ is the polar angle at time $t.$)
19. A particle moves in a plane perpendicular to the z-axis. The motion takes place along a circle with center on this axis. (a) Show that there is a vector $\pmb{\omega}(t)$ parallel to the $z$-axis such that $$ \mathbf{v}(t) = \pmb{\omega}(t) \times \mathbf{r}(t) $$ where $\mathbf{r}(t)$ and $\mathbf{v}(t)$ denote the position and velocity vectors at times $t.$ The vector $\pmb{\omega}(t)$ is called the angular velocity vector and its magnitude $\omega(t) = \|\pmb{\omega}(t)\|$ is called the angular speed. (b) The vector $\pmb{\alpha}(t) = \pmb{\omega}'(t)$ is called the angular acceleration vector. Show that the acceleration vector $\mathbf{a}(t) = \mathbf{v}'(t)$ is given by the formula \begin{align*} \\ \mathbf{a}(t) = \left[\pmb{\omega}(t) \cdot \mathbf{r}(t)\right]\pmb{\omega}(t) - \omega^2(t)\mathbf{r}(t) + \pmb{\alpha}(t) \times \mathbf{r}(t) \end{align*} (c) If the particle lies in the $xy$-plane and if the angular speed $\omega(t)$ is constant, say $\omega(t) = \omega,$ prove that the acceleration vector $\mathbf{a}(t)$ is centripetal and that, in fact, $\mathbf{a}(t) = -\omega^2\mathbf{r}(t).$
20. A body is said to undergo a rigid motion if, for every pair of particles $p$ and $q$ in the body, the distance $\|\mathbf{r}_p(t) - \mathbf{r}_q(t)\|$ is independent of $t,$ where $\mathbf{r}_p(t)$ and $\mathbf{r}_q(t)$ denote the position vectors of $p$ and $q$ at time $t.$ Prove that for a rigid motion in which each particle $p$ rotates about the $z$-axis we have $\mathbf{v}_p(t) = \pmb{\omega}(t) \times \mathbf{r}_p(t),$ where $\pmb{\omega}(t)$ is the same for each particle, and $\mathbf{v}_p(t)$ is the velocity of particle $p.$