$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$
-r(t) = (t + 10)i + (ln(cos t) - 6)j + 2k
A) a = (sec t tan t)T + (sec t)N
B) a = (sec t tan t)T - (sec t)N
C) a = (-sec t tan t)T + (sec t)N
D) a = (-sec t tan t)T - (sec t)N

Question

Quizplus · Accepted Answer

To find the acceleration in the form $a_{T}T + a_{N}N$, we first need to compute the acceleration vector $a(t)$ by differentiating the velocity vector $v(t)$, which is the derivative of the position vector $r(t)$. Given $r(t) = (t + 10)i + (\ln(\cos(t)) - 6)j + 2k$, we find $v(t)$ and then $a(t)$ by differentiation:1. Differentiate $r(t)$ to get $v(t)$:$$v(t) = \frac{d}{dt}[(t + 10)i + (\ln(\cos(t)) - 6)j + 2k] = (1)i + \left(-\tan(t)\right)j + 0k$$2. Differentiate $v(t)$ to get $a(t)$:$$a(t) = \frac{d}{dt}[(1)i + (-\tan(t))j + 0k] = 0i + (-\sec^2(t))j + 0k$$The acceleration vector $a(t)$ is purely in the direction of $j$, which means it's perpendicular to the $i$ component and has no component in the $k$ direction. Given the form of the acceleration vector, we can see that it has a component that is tangential (related to the change in speed) and a component that is normal (related to the change in direction).The correct answer must involve $\sec(t)$ and $\tan(t)$ due to the derivative of $\tan(t)$ being $\sec^2(t)$. The acceleration in the direction of $j$ (which is perpendicular to the direction of motion) contributes to the normal acceleration. Since the acceleration vector has a negative component in the $j$ direction and considering the trigonometric identities, the correct form involves $\sec(t)\tan(t)$ for the tangential component and $\sec(t)$ for the normal component, with the signs determined by the direction of these components.Thus, the correct answer is $a = (\sec(t)\tan(t))T + (\sec(t))N$, which matches choice A.

For the curve r(t), write the acceleration in the form aTT + anN. \text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. } For the curve r(t), write the acceleration in the form aTT + anN.

$\text { For the curve } r ( t ) \text {, write the acceleration in the form aTT } + \text { anN. }$