8.4: Motion in Space
- Page ID
- 144339
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = (3t^2 - 2, 2t - \sin t)\). Find the velocity, acceleration, and speed of this particle in terms of \(t\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = 2\sin t\,\mathbf{i} + 2\cos t\,\mathbf{j} + t^2\,\mathbf{k}\). Find the velocity, acceleration, and speed of the particle in terms of \(t\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = (e^{-t}, t^2, \tan t)\). Find the velocity, acceleration, and speed of the particle in terms of \(t\).
- A projectile is launched into the air from ground level with an initial speed of 50 m/sec at an angle of 60° with the horizontal. Construct a vector-valued function that models the position of the projectile \(t\) seconds after it is launched.
- A ball is thrown horizontally off the edge of a 540 m tall skyscraper with an initial speed of 40 m/sec. Construct a vector-valued function that models the position of the projectile \(t\) seconds after it is launched.
- A projectile is launched 1.5 m above the surface of Mars with an initial speed of 100 m/sec at an angle of 45° with the horizontal. Given that the acceleration due to gravity on Mars is \(g = -3.71 \text{m}/\text{sec}^2\), construct a vector-valued function that models the position of the projectile \(t\) seconds after it is launched.
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = \cos t\,\mathbf{i} + \sin t\,\mathbf{j} + t\,\mathbf{k}\). Find the tangential and normal components of acceleration when \(t = 0\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = \left(2t, t^2, \dfrac{t^3}{3}\right)\). Find the tangential and normal components of acceleration when \(t = 1\).
- The position of a particle at time \(t\) is given by the function \(\mathbf{r}(t) = 2t^3\,\mathbf{i} + 3t^2\,\mathbf{j} + 6t\,\mathbf{k}\). Find the tangential and normal components of acceleration at \(t = -1\).