Chapter 13

In Exercises \(11-14\) , find the arc length parameter along the curve from the point where \(t=0\) by evaluating the integral $$ s=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau $$ from Equation ( \(3 ) .\) Then find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad \pi / 2 \leq t \leq \pi $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k}\)

In Exercises \(9-14, \mathrm{r}(t)\) is the position of a particle in space at time \(t .\) Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of \(t .\) Write the particle's velocity at that time as the product of its speed and direction. $$ \mathbf{r}(t)=(\sec t) \mathbf{i}+(\tan t) \mathbf{j}+\frac{4}{3} t \mathbf{k}, \quad t=\pi / 6 $$

In Exercises \(11-14\) , find the arc length parameter along the curve from the point where \(t=0\) by evaluating the integral $$ s=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau $$ from Equation ( \(3 ) .\) Then find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+e^{t} \mathbf{k}, \quad-\ln 4 \leq t \leq 0 $$

In Exercises \(9-14, \mathrm{r}(t)\) is the position of a particle in space at time \(t .\) Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of \(t .\) Write the particle's velocity at that time as the product of its speed and direction. $$ \mathbf{r}(t)=(2 \ln (t+1)) \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad t=1 $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=\left(t^{3} / 3\right) \mathbf{i}+\left(t^{2} / 2\right) \mathbf{j}, \quad t>0\)

In Exercises \(11-14\) , find the arc length parameter along the curve from the point where \(t=0\) by evaluating the integral $$ s=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau $$ from Equation ( \(3 ) .\) Then find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(1+2 t) \mathbf{i}+(1+3 t) \mathbf{j}+(6-6 t) \mathbf{k}, \quad-1 \leq t \leq 0 $$

Suppose that \(\mathbf{r}\) is the position vector of a particle moving along a plane curve and \(d A / d t\) is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the necessary derivatives exist, give a geometric argument based on increments and limits for the validity of the equation $$ \frac{d A}{d t}=\frac{1}{2}|\mathbf{r} \times \dot{\mathbf{r}}| $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, \quad 0< t<\pi / 2\)

In Exercises \(11-14,\) write a in the form \(\mathbf{a}=a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) at the given value of \(t\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) $$ \mathbf{r}(t)=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+\sqrt{2 e^{t} \mathbf{k},} \quad t=0 $$

Equal-range firing angles Show that a projectile fired at an angle of \(\alpha\) degrees, \(0<\alpha<90\) , has the same range as a projectile fired at the same speed at an angle of \((90-\alpha)\) degrees. (In models that take air resistance into account, this symmetry is lost.)

In Exercises \(9-14, \mathrm{r}(t)\) is the position of a particle in space at time \(t .\) Find the particle's velocity and acceleration vectors. Then find the particle's speed and direction of motion at the given value of \(t .\) Write the particle's velocity at that time as the product of its speed and direction. $$ \mathbf{r}(t)=\left(e^{-t}\right) \mathbf{i}+(2 \cos 3 t) \mathbf{j}+(2 \sin 3 t) \mathbf{k}, \quad t=0 $$

Are length Find the length of the curve $$ \mathbf{r}(t)=(\sqrt{2} t) \mathbf{i}+(\sqrt{2} t) \mathbf{j}+\left(1-t^{2}\right) \mathbf{k} $$ from \((0,0,1)\) to \((\sqrt{2}, \sqrt{2}, 0)\)

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=t \mathbf{i}+(a \cosh (t / a)) \mathbf{j}, \quad a>0\)

In Exercises 15 and \(16,\) find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t .\) Then find equations for the osculating, normal, and rectifying planes at that value of \(t .\) $$ \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}-\mathbf{k}, \quad t=\pi / 4 $$

Equal-range firing angles What two angles of elevation will enable a projectile to reach a target 16 \(\mathrm{km}\) downrange on the same level as the gun if the projectile's initial speed is 400 \(\mathrm{m} / \mathrm{sec} ?\)

In Exercises \(15-18, \mathbf{r}(t)\) is the position of a particle in space at time \(t .\) Find the angle between the velocity and acceleration vectors at time \(t=0 .\) $$ \mathbf{r}(t)=(3 t+1) \mathbf{i}+\sqrt{3} t \mathbf{j}+t^{2} \mathbf{k} $$

In Exercises 16 and \(17,\) two planets, planet \(A\) and planet \(B\) , are orbiting their sun in circular orbits with \(A\) being the inner planet and \(B\) being farther away from the sun. Suppose the positions of \(A\) and \(B\) at time \(t\) are $$ \mathbf{r}_{A}(t)=2 \cos (2 \pi t) \mathbf{i}+2 \sin (2 \pi t) \mathbf{j} $$ and $$ \mathbf{r}_{B}(t)=3 \cos (\pi t) \mathbf{i}+3 \sin (\pi t) \mathbf{j} $$ respectively, where the sun is assumed to be located at the origin and distance is measured in astronomical units. (Notice that planet \(A\) moves faster than planet \(B . )\) The people on planet \(A\) regard their planet, not the sun, as the center of their planetary system (their solar system). Using planet \(A\) as the origin of a new coordinate system, give parametric equations for the location of planet \(B\) at time \(t .\) Write your answer in terms of \(\cos (\pi t)\) and \(\sin (\pi t) .\)

In Exercises 15 and \(16,\) find \(\mathbf{r}, \mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given value of \(t .\) Then find equations for the osculating, normal, and rectifying planes at that value of \(t .\) $$\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+t \mathbf{k}, \quad t=0$$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=(\cosh t) \mathbf{i}-(\sinh t) \mathbf{j}+t \mathbf{k}\)

Range and height versus speed a. Show that doubling a projectile's initial speed at a given launch angle multiplies its range by \(4 .\) b. By about what percentage should you increase the initial speed to double the height and range?

In Exercises \(15-18, \mathbf{r}(t)\) is the position of a particle in space at time \(t .\) Find the angle between the velocity and acceleration vectors at time \(t=0 .\) $$ \mathbf{r}(t)=\left(\frac{\sqrt{2}}{2} t\right) \mathbf{i}+\left(\frac{\sqrt{2}}{2} t-16 t^{2}\right) \mathbf{j} $$

Ellipse a. Show that the curve \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(1-\cos t) \mathbf{k}\) \(0 \leq t \leq 2 \pi,\) is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane. b. Sketch the ellipse on the cylinder. Add to your sketch the unit tangent vectors at \(t=0, \pi / 2, \pi,\) and \(3 \pi / 2 .\) c. Show that the acceleration vector always lies parallel to the plane (orthogonal to a vector normal to the plane). Thus, if you draw the acceleration as a vector attached to the ellipse, it will lie in the plane of the ellipse. Add the acceleration vectors for \(t=0, \pi / 2, \pi,\) and 3\(\pi / 2\) to your sketch. d. Write an integral for the length of the ellipse. Do not try to evaluate the integral; it is nonelementary. e. Numerical integrator Estimate the length of the ellipse to two decimal places.

In Exercises 16 and \(17,\) two planets, planet \(A\) and planet \(B\) , are orbiting their sun in circular orbits with \(A\) being the inner planet and \(B\) being farther away from the sun. Suppose the positions of \(A\) and \(B\) at time \(t\) are $$ \mathbf{r}_{A}(t)=2 \cos (2 \pi t) \mathbf{i}+2 \sin (2 \pi t) \mathbf{j} $$ and $$ \mathbf{r}_{B}(t)=3 \cos (\pi t) \mathbf{i}+3 \sin (\pi t) \mathbf{j} $$ respectively, where the sun is assumed to be located at the origin and distance is measured in astronomical units. (Notice that planet \(A\) moves faster than planet \(B . )\) The people on planet \(A\) regard their planet, not the sun, as the center of their planetary system (their solar system). Using planet \(A\) as the origin, graph the path of planet \(B .\) This exercise illustrates the difficulty that people before Kepler's time, with an earth- centered (planet \(A\) ) view of our solar system, had in understanding the motions of the planets (i.e., planet \(B=\) Mars). See D. G. Saari's article in the American Mathematical Monthly, Vol. 97 (Feb. \(1990 ),\) pp. \(105-119\) .

The speedometer on your car reads a steady 35 mph. Could you be accelerating? Explain.

Shot put In Moscow in 1987 , Natalya Lisouskaya set a women's world record by putting an 8 lb 13 oz shot 73 ft 10 in. Assuming that she launched the shot at a \(40^{\circ}\) angle to the horizontal from 6.5 \(\mathrm{ft}\) above the ground, what was the shot's initial speed?

In Exercises \(15-18, \mathbf{r}(t)\) is the position of a particle in space at time \(t .\) Find the angle between the velocity and acceleration vectors at time \(t=0 .\) $$ \mathbf{r}(t)=\left(\ln \left(t^{2}+1\right)\right) \mathbf{i}+\left(\tan ^{-1} t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k} $$

Show that the parabola \(y=a x^{2}, a \neq 0\) , has its largest curvature at its vertex and has no minimum curvature. (Note: since the curvature of a curve remains the same if the curve is translated or rotated, this result is true for any parabola.)

Length is independent of parametrization To illustrate that the length of a smooth space curve does not depend on the parametrization you use to compute it, calculate the length of one turn of the helix in Example 1 with the following parametrizations. $$ \begin{array}{l}{\text { a. } \mathbf{r}(t)=(\cos 4 t) \mathbf{i}+(\sin 4 t) \mathbf{j}+4 t \mathbf{k}, \quad 0 \leq t \leq \pi / 2} \\ {\text { b. } \mathbf{r}(t)=[\cos (t / 2)] \mathbf{i}+[\sin (t / 2)] \mathbf{j}+(t / 2) \mathbf{k}, \quad 0 \leq t \leq 4 \pi} \\ {\text { c. } \mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}-t \mathbf{k}, \quad-2 \pi \leq t \leq 0}\end{array} $$

Show that the ellipse \(x=a \cos t, y=b \sin t, a>b>0,\) has its largest curvature on its major axis and its smallest curvature on its minor axis. ( As in Exercise 17, the same is true for any ellipse.)

Can anything be said about the acceleration of a particle that is moving at a constant speed? Give reasons for your answer.

In Exercises \(15-18, \mathbf{r}(t)\) is the position of a particle in space at time \(t .\) Find the angle between the velocity and acceleration vectors at time \(t=0 .\) $$ \mathbf{r}(t)=\frac{4}{9}(1+t)^{3 / 2} \mathbf{i}+\frac{4}{9}(1-t)^{3 / 2} \mathbf{j}+\frac{1}{3} t \mathbf{k} $$

The involute of a circle If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end \(P\) traces an involute of the circle \(x^{2}+y^{2}=1\) and the tracing point circle in question is the circle \(x^{2}+y^{2}=1\) and the tracing point starts at \((1,0) .\) The unwound portion of the string is tangent to the circle at \(Q,\) and \(t\) is the radian measure of the angle from the positive \(x\) -axis to segment \(O Q .\) Derive the parametric equations $$ x=\cos t+t \sin t, \quad y=\sin t-t \cos t, \quad t>0 $$ of the point \(P(x, y)\) for the involute. Graph cannot copy

In Exercises 19 and \(20, \mathbf{r}(t)\) is the position vector of a particle in space at time \(t .\) Find the time or times in the given time interval when the velocity and acceleration vectors are orthogonal. $$ \mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi $$

In Example \(5,\) we found the curvature of the helix \(\mathbf{r}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}+b t \mathbf{k}\) \((a, b \geq 0)\) to be \(\kappa=a /\left(a^{2}+b^{2}\right) .\) What is the largest value \(\kappa\) can have for a given value of \(b ?\) Give reasons for your answer.

Can anything be said about the speed of a particle whose acceleration is always orthogonal to its velocity? Give reasons for your answer.

Firing from \(\left(x_{0}, y_{0}\right)\) Derive the equations $$ \begin{aligned} x &=x_{0}+\left(v_{0} \cos \alpha\right) t \\ y &=y_{0}+\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2} \end{aligned} $$ (see Equation ( 5\()\) in the text) by solving the following initial value problem for a vector \(\mathbf{r}\) in the plane. $$ \begin{array}{ll}{\text { Differential equation: }} & {\frac{d^{2} \mathbf{r}}{d t^{2}}=-g \mathbf{j}} \\ {\text { Initial conditions: }} & {\mathbf{r}(0)=x_{0} \mathbf{i}+y_{0} \mathbf{j}} \\ {} & {\frac{d \mathbf{r}}{d t}(0)=\left(v_{0} \cos \alpha\right) \mathbf{i}+\left(v_{0} \sin \alpha\right) \mathbf{j}}\end{array} $$

In Exercises 19 and \(20, \mathbf{r}(t)\) is the position vector of a particle in space at time \(t .\) Find the time or times in the given time interval when the velocity and acceleration vectors are orthogonal. $$ \mathbf{r}(t)=(\sin t) \mathbf{i}+t \mathbf{j}+(\cos t) \mathbf{k}, \quad t \geq 0 $$

An object of mass \(m\) travels along the parabola \(y=x^{2}\) with a constant speed of 10 units/ sec. What is the force on the object due to its acceleration at \((0,0) ?\) at \(\left(2^{1 / 2}, 2\right) ?\) Write your answers in terms of \(\mathbf{i}\) and \(\mathbf{j}\) . (Remember Newton's law, \(\mathbf{F}=m \mathbf{a} . )\)

Evaluate the integrals in Exercises \(21-26\) $$ \int_{0}^{1}\left[t^{3} \mathbf{i}+7 \mathbf{j}+(t+1) \mathbf{k}\right] d t $$

Find an equation for the circle of curvature of the curve \(\mathbf{r}(t)=t \mathbf{i}+(\sin t) \mathbf{j}\) at the point \((\pi / 2,1) .\) (The curve parametrizes the graph of \(y=\sin x\) in the \(x y\) -plane.)

Evaluate the integrals in Exercises \(21-26\) $$ \int_{1}^{2}\left[(6-6 t) \mathbf{i}+3 \sqrt{t} \mathbf{j}+\left(\frac{4}{t^{2}}\right) \mathbf{k}\right] d t $$

Find an equation for the circle of curvature of the curve \(\mathbf{r}(t)=\) \((2 \ln t) \mathbf{i}-[t+(1 / t)] \mathbf{j}, e^{-2} \leq t \leq e^{2},\) at the point \((0,-2)\) where \(t=1\)

Show that a moving particle will move in a straight line if the normal component of its acceleration is zero.

Evaluate the integrals in Exercises \(21-26\) $$ \int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t $$

Evaluate the integrals in Exercises \(21-26\) $$ \int_{0}^{\pi / 3}[(\sec t \tan t) \mathbf{i}+(\tan t) \mathbf{j}+(2 \sin t \cos t) \mathbf{k}] d t $$

Show that \(\kappa\) and \(\tau\) are both zero for the line $$ \mathbf{r}(t)=\left(x_{0}+A t\right) \mathbf{i}+\left(y_{0}+B t\right) \mathbf{j}+\left(z_{0}+C t\right) \mathbf{k} $$

Colliding marbles The figure shows an experiment with two marbles. Marble \(A\) was launched toward marble \(B\) with launch angle \(\alpha\) and initial speed \(v_{0} .\) At the same instant, marble \(B\) was released to fall from rest at \(R\) tan \(\alpha\) units directly above a spot \(R\) units downrange from \(A .\) The marbles were found to collide regardless of the value of \(v_{0} .\) Was this mere coincidence, or must this happen? Give reasons for your answer. Graph cannot copy

Evaluate the integrals in Exercises \(21-26\) $$ \int_{1}^{4}\left[\frac{1}{t} \mathbf{i}+\frac{1}{5-t} \mathbf{j}+\frac{1}{2 t} \mathbf{k}\right] d t $$

What can be said about the torsion of a smooth plane curve \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j} ?\) Give reasons for your answer.