Chapter 13

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}+\sqrt{5} t \mathbf{k}, \quad 0 \leq t \leq \pi $$

Period of Skylab 4 since the orbit of Skylab 4 had a semimajor axis of \(a=6808 \mathrm{km},\) Kepler's third law with \(M\) equal to Earth's mass should give the period. Calculate it. Compare your result with the value in Table \(13.2 .\)

For Exercises \(1-8\) you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) in Section 13.4 (Exercises \(9-16\) ). Find now \(\mathbf{B}\) and \(\tau\) for these space curves. $$ \mathbf{r}(t)=(3 \sin t) \mathbf{i}+(3 \cos t) \mathbf{j}+4 t \mathbf{k} $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves. \(\mathbf{r}(t)=t \mathbf{i}+(\ln \cos t) \mathbf{j}, \quad-\pi / 2< t< \pi / 2\)

Travel time A projectile is fired at a speed of 840 \(\mathrm{m} / \mathrm{sec}\) at an angle of \(60^{\circ} .\) How long will it take to get 21 \(\mathrm{km}\) downrange?

In Exercises \(1-4, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of \(t .\) $$ \mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=1 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k}, \quad 0 \leq t \leq \pi $$

In Exercises \(1-4, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of \(t .\) $$ \mathbf{r}(t)=\left(t^{2}+1\right) \mathbf{i}+(2 t-1) \mathbf{j}, \quad t=1 / 2 $$

Finding muzzle speed Find the muzzle speed of a gun whose maximum range is \(24.5 \mathrm{km} .\)

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=t \mathbf{i}+(2 / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq 8 $$

Flight time and height A projectile is fired with an initial speed of 500 \(\mathrm{m} / \mathrm{sec}\) at an angle of elevation of \(45^{\circ} .\) a. When and how far away will the projectile strike? b. How high overhead will the projectile be when it is 5 \(\mathrm{km}\) downrange? c. What is the greatest height reached by the projectile?

In Exercises \(1-4, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of \(t .\) $$ \mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2 t} \mathbf{j}, \quad t=\ln 3 $$

For Exercises \(1-8\) you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) in Section 13.4 (Exercises \(9-16\) ). Find now \(\mathbf{B}\) and \(\tau\) for these space curves. $$ \mathbf{r}(t)=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+2 \mathbf{k} $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves. \(\mathbf{r}(t)=(2 t+3) \mathbf{i}+\left(5-t^{2}\right) \mathbf{j}\)

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(2+t) \mathbf{i}-(t+1) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 3 $$

Semimajor axis of Viking \(\boldsymbol{I}\) The Viking \(I\) orbiter, which surveyed Mars from August 1975 to June 1976 , had a period of 1639 min. Use this and the mass of Mars, \(6.418 \times 10^{23} \mathrm{kg}\) , to find the semimajor axis of the Viking I orbit.

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the plane curves. \(\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad t>0\)

Throwing a baseball A baseball is thrown from the stands 32 \(\mathrm{ft}\) above the field at an angle of \(30^{\circ}\) up from the horizontal. When and how far away will the ball strike the ground if its initial speed is 32 \(\mathrm{ft} / \mathrm{sec} ?\)

In Exercises \(1-4, \mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle s velocity and acceleration vectors at the given value of \(t .\) $$ \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(3 \sin 2 t) \mathbf{j}, \quad t=0 $$

For Exercises \(1-8\) you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) in Section 13.4 (Exercises \(9-16\) ). Find now \(\mathbf{B}\) and \(\tau\) for these space curves. $$ \mathbf{r}(t)=(6 \sin 2 t) \mathbf{i}+(6 \cos 2 t) \mathbf{j}+5 t \mathbf{k} $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{j}+\left(\sin ^{3} t\right) \mathbf{k}, \quad 0 \leq t \leq \pi / 2 $$

For Exercises \(1-8\) you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) in Section 13.4 (Exercises \(9-16\) ). Find now \(\mathbf{B}\) and \(\tau\) for these space curves. $$ \mathbf{r}(t)=\left(t^{3} / 3\right) \mathbf{i}+\left(t^{2} / 2\right) \mathbf{j}, \quad t>0 $$

Shot put An athlete puts a \(16-16\) shot at an angle of \(45^{\circ}\) to the horizontal from 6.5 ft above the ground at an initial speed of 44 \(\mathrm{ft} / \mathrm{sec}\) as suggested in the accompanying figure. How long after launch and how far from the inner edge of the stopboard does the shot land? Graph cannot copy

Exercises \(5-8\) give the position vectors of particles moving along various curves in the \(x y-\) plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the circle \(x^{2}+y^{2}=1\) $$ \mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j} ; \quad t=\pi / 4 \text { and } \pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=6 t^{3} \mathbf{i}-2 t^{3} \mathbf{j}-3 t^{3} \mathbf{k}, \quad 1 \leq t \leq 2 $$

Period of Viking 2 The Viking 2 orbiter, which surveyed Mars from September 1975 to August \(1976,\) moved in an ellipse whose semimajor axis was \(22,030 \mathrm{km}\) . What was the orbital period? (Ex- press your answer in minutes.)

a. Show that the curvature of a smooth curve \(\mathbf{r}(t)=f(t) \mathbf{i}+\) \(g(t) \mathbf{j}\) defined by twice-differentiable functions \(x=f(t)\) and \(y=g(t)\) is given by the formula $$\kappa=\frac{|\dot{x} \ddot{y}-\dot{y} \ddot{x}|}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2}}$$ Apply the formula to find the curvatures of the following curves. b. \(\mathbf{r}(t)=t \mathbf{i}+(\ln \sin t) \mathbf{j}, \quad 0

Exercises \(5-8\) give the position vectors of particles moving along various curves in the \(x y-\) plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the circle \(x^{2}+y^{2}=16\) $$ \mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+(2 \sqrt{2} / 3) t^{3 / 2} \mathbf{k}, \quad 0 \leq t \leq \pi $$

Firing golf balls A spring gun at ground level fires a golf ball at an angle of \(45^{\circ} .\) The ball lands 10 \(\mathrm{m}\) away. a. What was the ball's initial speed? b. For the same initial speed, find the two firing angles that make the range 6 \(\mathrm{m}\) .

Exercises \(5-8\) give the position vectors of particles moving along various curves in the \(x y-\) plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the cycloid \(x=t-\sin t, y=1-\cos t\) $$ \mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2 $$

In Exercises \(1-8,\) find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. $$ \mathbf{r}(t)=(t \sin t+\cos t) \mathbf{i}+(t \cos t-\sin t) \mathbf{j}, \quad \sqrt{2} \leq t \leq 2 $$

The mass of Mars is \(6.418 \times 10^{23} \mathrm{kg}\) . If a satellite revolving about Mars is to hold a stationary orbit (have the same period as the period of Mars's rotation, which is 1477.4 min \()\) , what must the semimajor axis of its orbit be? Give reasons for your answer.

Exercises \(5-8\) give the position vectors of particles moving along various curves in the \(x y-\) plane. In each case, find the particle's velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. Motion on the parabola \(y=x^{2}+1\) $$ \mathbf{r}(t)=t \mathbf{i}+\left(t^{2}+1\right) \mathbf{j} ; \quad t=-1,0, \text { and } 1 $$

For Exercises \(1-8\) you found \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) in Section 13.4 (Exercises \(9-16\) ). Find now \(\mathbf{B}\) and \(\tau\) for these space curves. $$ \mathbf{r}(t)=(\cosh t) \mathbf{i}-(\sinh t) \mathbf{j}+t \mathbf{k} $$

Beaming electrons An electron in a TV tube is beamed horizontally at a speed of \(5 \times 10^{6} \mathrm{m} / \mathrm{sec}\) toward the face of the tube 40 \(\mathrm{cm}\) away. About how far will the electron drop before it hits?

Find the point on the curve $$ \mathbf{r}(t)=(5 \sin t) \mathbf{i}+(5 \cos t) \mathbf{j}+12 t \mathbf{k} $$ at a distance 26\(\pi\) units along the curve from the origin in the direction of increasing arc length.

In Exercises 9 and \(10,\) write a in the form \(a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) $$ \mathbf{r}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}+b t \mathbf{k} $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=(3 \sin t) \mathbf{i}+(3 \cos t) \mathbf{j}+4 t \mathbf{k}\)

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. $$ \mathbf{r}(t)=(t+1) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k}, \quad t=1 $$

Find the point on the curve $$ \mathbf{r}(t)=(12 \sin t) \mathbf{i}-(12 \cos t) \mathbf{j}+5 t \mathbf{k} $$ at a distance 13\(\pi\) units along the curve from the origin in the direction opposite to the direction of increasing are length.

In Exercises 9 and \(10,\) write a in the form \(a_{\mathrm{T}} \mathbf{T}+a_{\mathrm{N}} \mathbf{N}\) without finding \(\mathbf{T}\) and \(\mathbf{N} .\) $$ \mathbf{r}(t)=(1+3 t) \mathbf{i}+(t-2) \mathbf{j}-3 t \mathbf{k} $$

Find \(\mathbf{T}, \mathbf{N},\) and \(\kappa\) for the space curves. \(\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}+3 \mathbf{k}\)

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. $$ \mathbf{r}(t)=(1+t) \mathbf{i}+\frac{t^{2}}{\sqrt{2}} \mathbf{j}+\frac{t^{3}}{3} \mathbf{k}, \quad t=1 $$

Finding satellite speed A satellite moves around Earth in a circular orbit. Express the satellite's speed as a function of the orbit's radius.

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. $$ \mathbf{r}(t)=(4 \cos t) \mathbf{i}+(4 \sin t) \mathbf{j}+3 t \mathbf{k}, \quad 0 \leq t \leq \pi / 2 $$

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

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 \(\mathbf{N} .\) $$ \mathbf{r}(t)=(t+1) \mathbf{i}+2 t \mathbf{j}+t^{2} \mathbf{k}, \quad t=1 $$

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. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+4 t \mathbf{k}, \quad t=\pi / 2 $$

A golf ball leaves the ground at a \(30^{\circ}\) angle at a speed of \(90 \mathrm{ft} / \mathrm{sec} .\) Will it clear the top of a \(30-\mathrm{ft}\) tree that is in the way, 135 \(\mathrm{ft}\) down the fairway? Explain.