Exam 14: Vector-Valued Functions and Motion in Space

Calculate the arc length of the indicated portion of the curve r(t). - $r ( t ) = ( 7 t \sin t + 7 \cos t ) \mathbf { i } + ( 7 t \cos t - 7 \sin t ) \mathbf { j } ; - 3 \leq t \leq 5$

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 baseball is hit when it is $3.2 \mathrm { ft }$ above the ground. It leaves the bat with an initial velocity of $142 \mathrm { ft } / \mathrm { sec }$ at a launch angle of $30 ^ { \circ }$ . At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of $- 10.3 \mathrm { i } ( \mathrm { ft } / \mathrm { sec }$ ) to the ball's initial velocity. How high does the baseball go? Round your answer to the nearest tenth.

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the acceleration vector. r(t) = (cos 4t)i + (5 sin t)j

$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$ -r(t) = (5cos t)i + (5sin t)j + 2tk

Find the curvature of the curve r(t). - $\mathbf { r } ( \mathrm { t } ) = ( 3 + \ln ( \sec t ) ) \mathbf { i } + ( 1 + \mathrm { t } ) \mathbf { k } , - \pi / 2 < \mathrm { t } < \pi / 2$

Find the torsion of the space curve. - $\mathbf { r } ( \mathrm { t } ) = ( \mathrm { t } - 2 ) \mathbf { i } + ( \ln ( \cos \mathrm { t } ) - 4 ) \mathbf { j } + 6 \mathbf { k } , - \pi / 2 < \mathrm { t } < \pi / 2$

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the velocity vector. $\mathbf { r } ( \mathrm { t } ) = \left( - 8 \mathrm { t } ^ { 2 } - 3 \right) \mathbf { i } + \left( \frac { 1 } { 9 } \mathrm { t } ^ { 3 } \right) \mathbf { j }$

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector. -Find the velocity vector. r(t) = (cot t)i + (csc t)j

Solve the initial value problem. -Differential Equation: $\frac { \mathrm { d } \mathbf { r } } { \mathrm { dt } } = \left( \mathrm { t } ^ { 4 } + 5 \mathrm { t } ^ { 2 } \right) \mathbf { i } + 4 \mathrm { t } \mathbf { j }$ Initial Condition: $\mathbf { r } ( 0 ) = 5 \mathbf { i } - 6 \mathbf { j }$

Find the unit tangent vector of the given curve. - $r ( t ) = 3 t ^ { 5 } i - 12 t ^ { 5 } j + 4 t ^ { 5 } \mathbf { k }$

The position vector of a particle is r(t). Find the requested vector. - $\text { The acceleration at } t = 1 \text { for } r ( t ) = \left( 3 t - 3 t ^ { 4 } \right) \mathbf { i } + ( 3 - t ) \mathbf { j } + \left( 4 t ^ { 2 } - 4 t \right) k$

Find T, N, and B for the given space curve. - $\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 9 \right) \mathbf { i } + ( 2 \mathrm { t } - 9 ) \mathbf { j } + 4 \mathbf { k }$

$\text { Find the velocity vector in terms of } u _ { r } \text { and } u _ { \theta } \text {. }$ - $r = 2 \cos 2 t$ and $\theta = 4 t$

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 baseball is hit when it is $2.8 \mathrm { ft }$ above the ground. It leaves the bat with an initial velocity of $149 \mathrm { ft } / \mathrm { sec }$ at a launch angle of $24 ^ { \circ }$ . At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of $- 9.7 \mathrm { i } ( \mathrm { ft } / \mathrm { sec } )$ to the ball's initial velocity. Find a vector equation for the path of the baseball.

Find the unit tangent vector of the given curve. -r(t) = (7t cos t - 7 sin t)j + (7t sin t + 7 cos t)k

Provide an appropriate response. -A golf ball leaves the ground at a $25 ^ { \circ }$ angle at a speed of $88 \mathrm { ft } / \mathrm { sec }$ . Will it clear the top of a $31 \mathrm { ft }$ tree that is in the way, $135 \mathrm { ft }$ down the fairway? Explain.

$\text { Find the arc length parameter along the curve from the point where } t = 0 \text { by evaluating } s = \int _ { 0 } ^ { t } | v ( \tau ) | d \tau \text {. }$ - $\mathbf { r } ( \mathrm { t } ) = \left( e ^ { t } \cos t \right) \mathbf { i } + \left( e ^ { t } \sin t \right) \mathbf { j } + 5 e ^ { t } \mathbf { k }$

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$ - $\mathbf { r } ( \mathrm { t } ) = ( \mathrm { t } + 1 ) \mathbf { i } + ( \ln ( \sec t ) - 9 ) \mathbf { j } + 9 \mathbf { k } , - \pi / 2 < \mathrm { t } < \pi / 2$

Find T, N, and B for the given space curve. -r(t) = (cosh t)i + (sinh t)j + tk

The position vector of a particle is r(t). Find the requested vector. -The velocity at $t = \frac { \pi } { 4 }$ for $\mathbf { r } ( \mathrm { t } ) = 4 \sec ^ { 2 } ( \mathrm { t } ) \mathbf { i } - 5 \tan ( \mathrm { t } ) \mathbf { j } + 10 \mathrm { t } ^ { 2 } \mathbf { k }$