Mathematical Immaturity

14.9 Exercises

Exercises 1 through 6 below refer to the motions described in Exercises 1 through 6, respectively, of Section 14.7. For the value of $t$ specified:
(a) express the unit tangent $T$ and the principal normal $N$ in terms of $\mathbf{i}, \mathbf{j}, \mathbf{k}.$
(b) express the acceleration $\mathbf{a}$ as a linear combination of $T$ and $N.$

1. $t = 2$
2. $t = \pi$
3. $t = 0$
4. $t = \pi$
5. $t = 1$
6. $ t = \frac{\pi}{4} $
7. Prove that if the acceleration vector is always zero, the motion is linear.
8. Prove that the normal component of the acceleration vector is $\|\mathbf{v} \times \mathbf{a}\|/\|\mathbf{v}\|.$
9. For each of the following statements about a curve traced out by a particle moving in 3-space, either give a proof or exhibit a counter example.
   (a) If the velocity is constant, the curve lies in a plane.
   (b) If the speed is constant, the curve lies in a plane.
   (c) If the acceleration is constant, the curve lies in a plane.
   (d) If the velocity is perpendicular to the acceleration, the curve lies in a plane.
10. A particle of unit mass with position vector $\mathbf{r}(t)$ at time $t$ is moving in space under the actions of certain forces.
    (a) Prove that $\mathbf{r} \times \mathbf{r}'' = 0$ implies $\mathbf{r} \times \mathbf{r}' = \mathbf{c},$ where $\mathbf{c}$ is a constant vector.
    (b) If $\mathbf{r} \times \mathbf{r}' = \mathbf{c},$ where $\mathbf{c}$ is a constant vector, prove that the motion takes place in a plane. Consider both $\mathbf{c} \neq 0$ and $\mathbf{c} = 0.$
    (c) If the net force acting on the particle is always directed toward the origin, prove that the particle moves in a plane.
    (d) Is $\mathbf{r} \times \mathbf{r}'$ necessarily constant if a particle moves in a plane?
11. A particle moves along a curve in such a way that the velocity vector makes a constant angle with a given unit vector $\mathbf{c}.$
    (a) If the curve lies in a plane containing $\mathbf{c},$ prove that the acceleration vector is either zero or parallel to the velocity.
    (b) Give an example of such a curve (not a plane curve) for which the acceleration vector is never zero nor parallel to the velocity.
12. A particle moves along the ellipse $3x^2 + y^2 = 1$ with position vector $$ \begin{align*} \mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} \end{align*} $$ The motion is such that the horizontal component of the velocity vector at time $t$ is $-g(t).$
    (a) Does the particle move around the ellipse in a clockwise or counterclockwise direction?
    (b) Prove that the vertical component of the velocity vector at time $t$ is proportional to $f(t)$ and find the factor of proportionality.
    (c) How much time is required for the particle to go once around the ellipse?
13. A plane curve $C$ in the first quadrant has a negative slope at each of its points and passes through the point $(\frac{3}{2}, 1).$ The position vector $\mathbf{r}$ from the origin to any point $(x,y)$ on $C$ makes an angle $\theta$ with $\mathbf{i},$ and the velocity vector makes an angle $\phi$ with $\mathbf{i},$ where $0 \lt \theta \lt \frac{\pi}{2},$ and $0 \lt \phi \lt \frac{\pi}{2}.$ If $3 \tan \phi = 4 \cot \theta$ at each point of $C,$ find a Cartesian equation for $C$ and sketch the curve.
14. A line perpendicular to the tangent line of a plane curve is called a normal line. If the normal line and a vertical line are drawn at any point of a certain plane curve $C,$ they cut off a segment of length 2 on the x-axis. Find a Cartesian equation for this curve if it passes through the point $(1,2).$ Two solutions are possible.
15. Given two fixed nonzero vectors $A$ and $B$ making an angle $\theta$ with each other, where $0 < \theta < \pi.$ A motion with position vector $\mathbf{r}(t)$ at time $t$ satisfies the differential equation $\mathbf{r}'(t) = A \times \mathbf{r}(t)$ and the initial condition $\mathbf{r}(0) = B.$ (a) Prove that the acceleration $\mathbf{a}(t)$ is orthogonal to $A.$ (b) Prove that the speed is constant and compute this speed in terms of $A,$ $B,$ and $\theta.$ (c) Make a sketch of the curve, showing its relation to the vectors $A$ and $B.$
16. This exercise describes how the unit tangent and the principal normal are affected by a change of parameter. Suppose a curve $C$ is described by two equivalent functions $X$ and $Y,$ where $Y(t) = X[u(t)].$ Denote the unit tangent for $X$ by $T_X$ and that for $Y$ by $T_Y.$ (a) Prove that at each point of $C$ we have $T_Y(t) = T_X[u(t)]$ if $u$ is strictly increasing, but that $T_Y(t) = -T_X[u(t)]$ if $u$ is strictly decreasing. In the first case, $u$ is said to preserve orientation; in the second case, $u$ is said to reverse orientation. (b) Prove that the corresponding principal normal vectors $N_X$ and $N_Y$ satisfy $N_Y(t) = N_X[u(t)]$ at each point of $C.$ Deduce that the osculating plane is invariant under a change of parameter.