- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.7 Exercises
- In each of Exercises 1 through 6, $\textbf{r}(t)$ denotes the position vector at time $t$ for a particle moving on a space curve. Determine the velocity $\textbf{v}(t)$ and acceleration $\textbf{a}(t)$ in terms of $\mathbf{i}, \mathbf{j}, \mathbf{k};$ also, compute the speed $v(t).$ $$ \begin{align*} \\ \textbf{r}(t) = (t - \sin t)\mathbf{i} + (1 - \cos t)\mathbf{j} + 4\sin \frac{t}{2}\mathbf{k} \end{align*} $$
- $$ \begin{align*} \textbf{v}(t) &= (1 - \cos t)\,\mathbf{i} + \sin t\,\mathbf{j} + 2\cos \frac{t}{2}\,\mathbf{k} \\ \\ \textbf{a}(t) &= \sin t\,\mathbf{i} + \cos t\,\mathbf{j} - \sin \frac{t}{2}\,\mathbf{k} \\ \\ v(t) &= \sqrt{1 - 2 \cos t + \cos^2 t + \sin^2 t + 4 \cos^2 \frac{t}{2}} \\ \\ &= 2 \end{align*} $$