Mathematical Immaturity

14.7 Exercises

In each of Exercises 1 through 6, $\textbf{r}(t)$ denotes the position vector at time $t$ for a particle moving on a space curve. Determine the velocity $\textbf{v}(t)$ and acceleration $\textbf{a}(t)$ in terms of $\mathbf{i}, \mathbf{j}, \mathbf{k};$ also, compute the speed $v(t).$

1. $\textbf{r}(t) = (3t - t^3)\,\mathbf{i} + 3t^3\,\mathbf{j} + (3t + t^3)\,\mathbf{k}$
2. $ \textbf{r}(t) = \cos t\,\mathbf{i} + \sin t\,\mathbf{j} + e^t\,\mathbf{k} $
3. $ \textbf{r}(t) = 3t\cos t\mathbf{i} + 3t\sin t\mathbf{j} + 4t\,\mathbf{k} $
4. $ \textbf{r}(t) = (t - \sin t)\mathbf{i} + (1 - \cos t)\mathbf{j} + 4\sin \frac{t}{2}\mathbf{k} $
5. $ \textbf{r}(t) = 3t^2\mathbf{i} + 2t^3\mathbf{j} + 3t\mathbf{k} $
6. $ \textbf{r}(t) = t\mathbf{i} + \sin t\mathbf{j} + (1 - \cos t)\,\mathbf{k} $
7. Consider the helix described by the vector equation $\textbf{r}(t) = a\cos \omega t\mathbf{i} + a\sin \omega t\mathbf{j} + b\omega t\,\mathbf{k},$ where $\omega$ is a positive constant. Prove that the tangent line makes a constant angle with the $z$-axis and that the cosine of this angle is $b/\sqrt{a^2 + b^2}.$
8. Referring to the helix in Exercise 7, prove that the velocity $v$ and acceleration $a$ are vectors of constant length, and that: $$\frac{\|\mathbf{v} \times \mathbf{a}\|}{\|\mathbf{v}\|^3} = \frac{a}{a^2 + b^2}$$
9. Referring to Exercise 7, let $\mathbf{u}(t)$ denote the unit vector $\mathbf{u}(t) = \sin \omega t\mathbf{i} - \cos \omega t\mathbf{j}.$ Prove that there are constants $A$ and $B$ such that $\mathbf{v} \times \mathbf{a} = A\mathbf{u}(t) + B\,\mathbf{k},$ and express $A$ and $B$ in terms of $a,$ $b,$ and $\omega.$
10. Prove that for any motion the dot product of the velocity and acceleration is half the derivative of the square of the speed: $$\textbf{v}(t) \cdot \textbf{a}(t) = \frac{1}{2}\frac{d}{dt}v^2(t)$$
11. Let $\mathbf{c}$ be a fixed unit vector. A particle moves in space in such a way that its position vector $\textbf{r}(t)$ satisfies the equation $\textbf{r}(t) \cdot \mathbf{c} = e^{2t}$ for all $t,$ and its velocity vector $\textbf{v}(t)$ makes a constant angle $\theta$ with $\mathbf{c},$ where $0 < \theta < \frac{\pi}{2}.$ (a) Prove that the speed at time $t$ is $2e^{2t}/\cos \theta.$ (b) Compute the dot product $\textbf{a}(t) \cdot \textbf{v}(t)$ in terms of $t$ and $\theta.$
12. The identity $\cosh^2 \theta - \sinh^2 \theta = 1$ for hyperbolic functions suggests that the hyperbola $x^2/a^2 - y^2/b^2 = 1$ may be represented by the parametric equations $x = a\cosh \theta,$ $y = b\sinh \theta,$ or what amounts to the same thing, by the vector equation $r = a\cosh \theta\mathbf{i} + b\sinh \theta\mathbf{j}.$ When $a = b = 1,$ the parameter $\theta$ may be given a geometric interpretation analogous to that which holds between $\theta,$ $\sin \theta,$ and $\cos \theta$ in the unit circle shown in Figure 14.7(a). Figure 14.7(b) shows one branch of the hyperbola $x^2 - y^2 = 1.$ If the point $P$ has coordinates $x = \cosh \theta$ and $y = \sinh \theta,$ prove that $\theta$ equals twice the area of the sector $OAP$ shaded in the figure. [Hint: Let $A(\theta)$ denote the area of sector $OAP.$ Show that $$ \begin{align*} \\ A(\theta) = \frac{1}{2}\cosh \theta \sinh \theta - \int_1^{\cosh \theta} \sqrt{x^2 - 1}\,dx \end{align*} $$ Differentiate to get $A'(\theta) = \frac{1}{2}.$]
13. A particle moves along a hyperbola according to the equation $$\textbf{r}(t) = a\cosh \omega t\,\mathbf{i} + b\sinh \omega t\,\mathbf{j}$$ where $\omega$ is a constant. Prove that the acceleration is centrifugal.
14. Prove that the tangent line at a point $X$ of a parabola bisects the angle between the line joining $X$ to the focus and the line through $X$ parallel to the axis. This gives the reflection property of the parabola. (See Figure 14.3.)
15. A particle of mass 1 moves in a plane according to the equation $\textbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j}.$ It is attracted toward the origin by a force whose magnitude is four times its distance from the origin. At time $t = 0,$ the initial position is $r(0) = 4\mathbf{i}$ and the initial velocity is $v(0) = 6\mathbf{j}.$ (a) Determine the components $x(t)$ and $y(t)$ explicitly in terms of $t.$ (b) The path of the particle is a conic section. Find a Cartesian equation for this conic, sketch the conic, and indicate the direction of motion along the curve.
16. A particle moves along the parabola $x^2 + c(y - x) = 0$ in such a way that the horizontal and vertical components of the acceleration vector are equal. If it takes $T$ units of time to go from the point $(c, 0)$ to the point $(0, 0),$ how much time will it require to go from $(c, 0)$ to the halfway point $(c/2, c/4)$?
17. Suppose a curve $C$ is described by two equivalent functions $X$ and $Y,$ where $Y(t) = X[u(t)].$ Prove that at each point of $C$ the velocity vectors associated with $X$ and $Y$ are parallel, but that the corresponding acceleration vectors need not be parallel.