Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 14 - Calculus of Vector-Valued Functions - 14.5 Motion in 3-Space - Preliminary Questions - Page 744: 4


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Work Step by Step

We have $$ \mathbf{a}(t)=a_{\mathbf{T}}(t) \mathbf{T}(t)+a_{\mathbf{N}}(t) \mathbf{N}(t) $$ where $a_{\mathbf{T}}(t) $ is the tangential component of the accelaration and $a_{\mathbf{N}}(t) $ is the normal one. Now, if the speed is constant, then $v'(t)=0$ and hence $a_{\mathbf{T}}(t) =0$. So $$ \mathbf{a}(t)=a_{\mathbf{N}}(t) \mathbf{N}(t) $$ Thus, the acceleration is in the direction of the normal vector, which is orthogonal to the velocity vector. That is, the acceleration and velocity vectors are orthogonal.
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