- 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
- A particle moves along a hyperbola according to the equation $$\textbf{r}(t) = a\cosh \omega t\,\mathbf{i} + b\sinh \omega t\,\mathbf{j}$$ where $\omega$ is a constant. Prove that the acceleration is centrifugal.
- By definition, acceleration is centrifugal if the position vector and acceleration vector have the same direction (ie, away from the center of the hyperbola at the origin).
- We wish to show that for some positive scalar $c,$ that $\mathbf{a} = c\,\mathbf{r}.$ Differentiating with respect to $t,$ we get \begin{align*} \mathbf{v}(t) &= \omega a\,\sinh \omega t\,\mathbf{i} + \omega b\,\cosh \omega t\,\mathbf{j} \\ \mathbf{a}(t) &= \omega^2 a\,\cosh \omega t\,\mathbf{i} + \omega^2 b\,\sinh \omega t\,\mathbf{j} \\ &= \omega^2\, \mathbf{r}{(t)} \quad \blacksquare \end{align*}