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Exam 14: Vector-Valued Functions and Motion in Space

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Question 61

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Provide an appropriate response. -The following equations each describe the motion of a particle. For which path is the particle's speed constant? (1) $r ( t ) = t ^ { 3 } i + t ^ { 2 } j$ (2) $r ( t ) = \cos ( 8 t ) i + \sin ( 6 t ) j$ (3) $r ( t ) = t i + t j$ (4) $r ( t ) = \cos \left( - 5 t ^ { 2 } \right) \mathbf { i } + \sin \left( - 5 t ^ { 2 } \right) \mathbf { j }$

A) Path (4) and Path (2)
B) Path (1)
C) Path (2) and Path (3)
D) Path (3)

E) B) and C)
F) A) and B)

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Question 62

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For the curve r(t) , find an equation for the indicated plane at the given value of t. - $r ( t ) = \frac { 40 } { 9 } ( 1 + t ) ^ { 3 / 2 } \mathbf { i } + \frac { 40 } { 9 } ( 1 - t ) ^ { 3 / 2 } \mathbf { j } + \frac { 10 } { 3 } t \mathbf { k }$ ; osculating plane $a t \mathrm { t } = 0$

A) $x - y = 0$
B) $x - y = - \frac { 40 } { 9 }$
C) $x - y = \frac { 40 } { 9 }$
D) $x - y = \frac { 80 } { 9 }$

E) B) and D)
F) A) and D)

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Question 63

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Find the curvature of the space curve. - $\mathbf { r } ( \mathrm { t } ) = 5 \mathrm { ti } + \left( 4 + 8 \cos \frac { 3 } { 2 } \mathrm { t } \right) \mathbf { j } + \left( 4 + 8 \sin \frac { 3 } { 2 } \right) \mathbf { k }$

Find the curvature of the curve r(t) . -r(t) = (8 + cos 9t - sin 9t) i + (5 + sin 9t + cos 9t) j + 7k

Find the unit tangent vector of the given curve. - $r ( t ) = \left( 5 \sin ^ { 3 } 9 t \right) i + \left( 5 \cos ^ { 3 } 9 t \right) \mathbf { j }$

The position vector of a particle is r(t) . Find the requested vector. -The velocity at $t = 0$ for $\mathbf { r } ( \mathrm { t } ) = \ln \left( \mathrm { t } ^ { 3 } - 3 \mathrm { t } ^ { 2 } + 4 \right) \mathbf { i } - \sqrt { 2 \mathrm { t } + 36 } \mathbf { j } - 5 \cos ( \mathrm { t } ) \mathbf { k }$

Provide an appropriate response. -A human cannonball is to be fired with an initial speed of $80 \mathrm { ft } / \mathrm { sec }$ . The circus performer hopes to land on a cushion located 190 feet downrange at the same height as the muzzle of the cannon. The circus is being held in a large room with a flat ceiling 33 feet higher than the muzzle. Can the performer be fired to the cushion without striking the ceiling? If so, what is the proper firing angle? ( $\mathrm { g } = 32 \mathrm { ft } / \sec ^ { 2 }$ )

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } = ( 8 \mathrm { t } - 3 ) \mathbf { i }$ Initial Conditions: $\left. \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } \right| _ { \mathrm { r } = 0 } = - \mathbf { k } , \mathrm { r } ( 0 ) = 8 \mathbf { i } + 7 \mathbf { j } + 6 \mathbf { k }$

Find the principal unit normal vector N for the curve r(t) . -r(t) = (10 sin t) i + (10 cos t) j + 9k

Solve the problem. -Show that a planet in a circular orbit moves with constant speed.

Calculate the arc length of the indicated portion of the curve r(t) . -Following the curve $r ( t ) = 5 t i + ( 12 \cos t ) j + ( 12 \sin t ) k$ in the direction of increasing of arc length, find the point that lies $26 \pi$ units away from the point where $t = 0$ .

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface -A projectile is fired at a speed of $720 \mathrm {~m} / \mathrm { sec }$ at an angle of $45 ^ { \circ }$ . How long will it take to get $18 \mathrm {~km}$ downrange? Round your answer to the nearest whole number.

Provide an appropriate response. -Derive the equations $\begin{array} { l } x = x _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \cos \alpha \\y = y _ { 0 } + \frac { v _ { 0 } } { k } \left( 1 - e ^ { - k t } \right) \sin \alpha + \frac { g } { k ^ { 2 } } \left( 1 - k t - e ^ { - k t } \right)\end{array}$ by solving the following initial value problem for a vector $\mathbf { r }$ in the plane. $\begin{aligned}\text { Differential equation } \frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } & = - g \mathbf { j } - \mathrm { k } \mathbf { v } = - \mathrm { g } \mathbf { j } - \mathrm { k } \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } \\\text { Initial conditions: } \quad \mathrm { r } ( 0 ) & = \mathrm { x } _ { 0 } \mathbf { i } + \mathrm { y } _ { 0 } \mathbf { j } \\& \frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } ( 0 ) = \mathbf { v } _ { 0 } = \left( \mathrm { v } _ { 0 } \cos \alpha \right) \mathbf { i } + \left( \mathrm { v } _ { 0 } \sin \alpha \right) \mathbf { j }\end{aligned}$ The drag coefficient $\mathrm { k }$ is a positive constant representing resistance due to air density, vo and $\alpha$ are the projectile's initial speed and launch angle, and $g$ is the acceleration of gravity.

Solve the problem. -The orbit of a satellite had a semimajor axis of $a = 6955 \mathrm {~km}$ . Calculate the period of the satellite. (Earth's mass $= 5.975 \times 10 ^ { 24 } \mathrm {~kg}$ and $\mathrm { G } = 6.6720 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} - 2$ ) .

For the curve r(t) , find an equation for the indicated plane at the given value of t. - $\left. \mathbf { r } ( \mathrm { t } ) = \left( 3 \sin \frac { 5 } { 3 } \mathrm { t } + 7 \right) \mathbf { i } + ( 3 \cos 20 \mathrm { t } ) - 6 \right) \mathbf { j } + 12 \mathrm { k } \mathbf { k }$ ; osculating plane at $t = 2.5 \pi$ .

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ -r(t) = (6t sin t + 6 cos t) i + (6t cos t - 6 sin t) j + 3k

For the curve r(t) , find an equation for the indicated plane at the given value of t. - $\mathbf { r } ( \mathrm { t } ) = ( \mathrm { t } + 6 ) \mathbf { i } + ( \ln ( \cos \mathrm { t } ) - 8 ) \mathbf { j } + 7 \mathbf { k } , - \pi / 2 < \mathrm { t } < \pi \cdot 2 ;$ rectifying plane $\mathrm { at } \mathrm {} \mathrm { t } = \frac { \pi } { 4 }$ .

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ -r(t) = (t + 10) i + (ln(cos t) - 6) j + 2k

Calculate the arc length of the indicated portion of the curve r(t) . - $r ( t ) = \left( 2 \cos ^ { 3 } 5 t \right) \mathbf { j } + \left( 2 \sin ^ { 3 } 5 t \right) \mathbf { k } ; \frac { 3 } { 5 } \pi \leq t \leq \frac { 7 } { 10 } \pi$

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured from the horizontal, and that the projectile is launched from the origin over a horizontal surface -An athlete puts a 16-lb shot at an angle of $39 ^ { \circ }$ to the horizontal from $6.1 \mathrm { ft }$ above the ground at an initial speed of $47 \mathrm { ft } / \mathrm { sec }$ . How far forward does the shot travel before it hits the ground? Round your answer to the nearest tenth.