- 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
-
A particle moves along a curve in such a way that the velocity vector makes a constant angle with a given unit vector $\mathbf{c}.$
(a) If the curve lies in a plane containing $\mathbf{c},$ prove that the acceleration vector is either zero or parallel to the velocity.
(b) Give an example of such a curve (not a plane curve) for which the acceleration vector is never zero nor parallel to the velocity.
-
(a) Break down acceleration into its tangential and normal components
$$
\mathbf{a}(t) = v'(t)T(t) + v(t)\|T'(t)\|N(t)
$$
If the velocity vector makes a constant angle with $\mathbf{c},$ what does this imply about $T$ and $T'?$
-
(a) If the velocity vector makes a constant angle with $\mathbf{c},$ then this implies that $T$ is constant. But that means that $T'$ must be zero, thus making the normal component of acceleration zero. Then, acceleration is
$$
\mathbf{a}(t) = v'(t)T(t)
$$
If $v'(t) = 0,$ then $\mathbf{a}(t) = 0.$ Otherwise, acceleration is a scalar multiple of $T,$ making it parallel to velocity.
