Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 13 - Section 13.4 - Motion in Space: Velocity and Acceleration - 13.4 Exercise - Page 879: 38



Work Step by Step

Formula to calculate the tangential acceleration component is: $a_T=\dfrac{r'(t) \cdot r''(t)}{|r'(t)|}$ ....(1) and Formula to calculate the normal acceleration component is: $a_N=\dfrac{r'(t) \times r''(t)}{|r'(t)|}$ ....(2) Thus, $r'(t)= 4ti+(2t^2-2) j$ and $r''(t)=4i+4tj$ $|r'(t)|=\sqrt{(4t)^2+(2t^2-2)}=2(t^2+1)$ $r'(t) \cdot r''(t)=[4ti+(2t^2-2) j] \cdot [4i+4tj]=8t^3+8t$ and $r'(t) \times r''(t)=[4ti+(2t^2-2) j] \times [4i+4tj]=(8t^2+8)k$ From equation (1), we have $a_T=\dfrac{r'(t) \cdot r''(t)}{|r'(t)|}=\dfrac{8t^3+8t}{2(t^2+1)}=4t$ From equation (2), we have $a_N=\dfrac{|r'(t) \times r''(t)|}{|r'(t)|}=\dfrac{(8t^2+8)}{2(t^2+1)}=4$
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