- 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
- Prove that for any motion the dot product of the velocity and acceleration is half the derivative of the square of the speed: $$\textbf{v}(t) \cdot \textbf{a}(t) = \frac{1}{2}\frac{d}{dt}v^2(t)$$
- Recall that $v^2(t) = \mathbf{v}(t) \cdot \mathbf{v}(t).$ Then, apply the dot product rule for vector derivatives.
- $$\displaylines{\mathbf{v}(t) \cdot \mathbf{a}(t)\ = \frac{1}{2}\left(\mathbf{v}(t) \cdot \mathbf{v}(t)\right)'\\ = \frac{1}{2}\frac{d}{dt}v^2(t) \quad \blacksquare}$$