- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.9 Exercises
- Prove that if the acceleration vector is always zero, the motion is linear.
- If $\mathbf{a}(t) = 0$ for all $t,$ then this means that $\mathbf{v}(t)$ is constant for all $t.$ This means that the speed v(t) is also constant, making v'(t) = 0, thus making the formula for acceleration $$ \begin{align*} \mathbf{a}(t) &= v(t)T'(t)\\ &= O \end{align*} $$ In the trivial case where $v(t) =0$ for all $t,$ then there is no motion. If acceleration is zero for all $t$ with nonzero speed, then $T'(t)$ must be zero for all $t.$ And since $T'(t)$ is the measure of change in direction of $T(t),$ this means that there is no change in direction at any time $t,$ thus the motion must be linear.