Exam 14: Vector-Valued Functions and Motion in Space

$\text { Find the velocity vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$ - $\mathrm { r } = \mathrm { e } ^ { \mathrm { a } \theta } \text { and } \frac { \mathrm { d } \theta } { \mathrm { dt } } = 3$

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 ideal projectile is launched from level ground at a launch angle of $26 ^ { \circ }$ and an initial speed of $48 \mathrm {~m} / \mathrm { sec }$ . How far away from the launch point does the projectile hit the ground?

Provide an appropriate response. -What two angles of elevation will enable a projectile to reach a target 17 km downrange on the same level as the gun if the projectile's initial speed is 420 m/sec? Assume there is no wind resistance.

Evaluate the integral. - $\int _ { 0 } ^ { 3 } \left[ \left( \frac { 3 } { \sqrt { 1 + t } } \right) i - \left( 5 t ^ { 2 } \right) j + \left( \frac { 12 t } { \left( 1 + t ^ { 2 } \right) ^ { 2 } } \right) k \right] d t$

For the curve r(t), find an equation for the indicated plane at the given value of t. - $\mathbf { r } ( \mathrm { t } ) = ( \mathrm { t } - 9 ) \mathbf { i } + ( \ln ( \sec \mathrm { t } ) - 1 ) \mathbf { j } + 7 \mathbf { k } , - \pi / 2 < \mathrm { t } < \pi / 2$ ; normal plane at $t = 10 \pi$ .

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 -A projectile is fired with an initial speed of $512 \mathrm {~m} / \mathrm { sec }$ at an angle of $45 ^ { \circ }$ . What is the greatest height reached by the projectile? Round your answer to the nearest tenth.

Calculate the arc length of the indicated portion of the curve r(t). - $r ( t ) = 6 \sqrt { 2 } t ^ { 3 / 2 } \mathbf { i } + ( 9 t \sin t ) \mathbf { j } + ( 9 t \cos t ) \mathbf { k } ; - 3 \leq t \leq 7$

Provide an appropriate response. - $\text { Prove that if } \mathbf { v } \text { and } \mathbf { u } \text { are differentiable functions of } t \text {, then } \frac { \mathrm { d } } { \mathrm { dt } } ( \mathbf { u } + \mathbf { v } ) = \frac { \mathrm { d } \mathbf { u } } { \mathrm { dt } } + \frac { \mathrm { d } \mathbf { v } } { \mathrm { dt } } \text {. }$

Find the unit tangent vector of the given curve. - $r ( t ) = ( 2 \cos 6 t ) i + ( 2 \sin 6 t ) j - 5 t k$

The position vector of a particle is r(t). Find the requested vector. -The acceleration at $t = \frac { \pi } { 4 }$ for $r ( t ) = ( 2 \sin 2 t ) \mathbf { i } - ( 2 \cos 2 t ) \mathbf { j } + ( 5 \csc 2 t ) \mathbf { k }$

Solve the problem. -At time t = 0 a particle is located at the point (4, -3, 2). It travels in a straight line to the point (6, -2, 1), has speed 2 at (4, -3, 2) and constant acceleration 6i - j - k. Find an equation for the position vector r(t) of the particle at time t.

Solve the initial value problem. -Differential Equation: $\frac { d \mathbf { r } } { d t } = \mathbf { i } + \left( 4 t ^ { 3 } - 10 t \right) j + \frac { 1 } { t + 2 } \mathbf { k }$ Initial Condition: $\mathbf { r } ( 0 ) = 2 \mathrm { j } + ( \ln 2 ) \mathbf { k }$

Provide an appropriate response. -The following equations each describe the motion of a particle. For which path is the particle's velocity vector alv orthogonal to its acceleration vector? (1) $r ( t ) = t ^ { 5 } i + t ^ { 3 } j$ (2) $r ( t ) = \cos ( 10 t ) \mathbf { i } + \sin ( 9 t ) \mathbf { j }$ (3) $r ( t ) = t i + t ^ { 3 } j$ (4) $r ( t ) = \cos ( 4 t ) i + \sin ( 4 t ) j$

Provide an appropriate response. -Derive the equations x=+ \alpha t y=+ \alpha t- by solving the following initial value problem for a vector $\mathbf { r }$ in the plane. Differential equation $\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } = - \mathrm { gj }$ Initial conditions: (0)=+ (0)=\alpha + \alpha

Find T, N, and B for the given space curve. - $r ( t ) = \left( 8 + 6 \sin \frac { 2 } { 3 } t \right) i + \left( 4 + 6 \cos \frac { 2 } { 3 } t \right) j + 3 t k$

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } = \left( \sec ^ { 2 } \mathrm { t } \right) \mathbf { i } + \left( 4 \mathrm { t } ^ { 3 } - 6 \right) \mathbf { j }$ Initial Condition: $\mathbf { r } ( 0 ) = - 4 \mathbf { j }$

$\text { For the smooth curve } r ( t ) \text {, find the parametric equations for the line that is tangent to } r \text { at the given parameter value } t = t _ { 0 } \text {. }$ - $\mathbf { r } ( \mathrm { t } ) = ( 9 \tan \mathrm { t } ) \mathbf { i } - ( 3 \sin \mathrm { t } ) \mathbf { j } + \left( 10 \cos ^ { 2 } \mathrm { t } \right) \mathbf { k } ; \mathrm { t } _ { 0 } = \frac { \pi } { 4 }$

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm { dt } ^ { 2 } } = 3 \mathrm { t } ^ { 2 } \mathbf { i } - \mathrm { tj }$ Initial Conditions: $\left. \frac { \mathrm { d } r } { \mathrm { dt } } \right| _ { \mathrm { r } = 0 } = - 4 \mathbf { i } , \mathrm { r } ( 0 ) = 10 \mathbf { i } - 3 \mathbf { k }$

Find the principal unit normal vector N for the curve r(t). -r(t) = (2t sin t + 2 cos t)i + (2t cos t - 2 sin t))k

Solve the problem. -A space shuttle is in a circular orbit $250 \mathrm {~km}$ above the Earth's surface. Use Kepler's third law (with $\mathrm { a } =$ Earth's radius $+ 250 \mathrm {~km} )$ to find the orbital period of the satellite. $\left( \mathrm { G } = 6.67 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 } \right.$ , $\mathrm { M } = 5.97 \times 10 ^ { 24 } \mathrm {~kg}$ and Earth's radius is $6.37 \times 10 ^ { 6 } \mathrm {~m}$ ).