Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } ( \cos 8 t , \sin 8 t , 8 )$
A) $a _ { \mathbf { T } } = \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = 8 \sqrt { 65 } e ^ { t }$
B) $a _ { \mathbf { T } } = 8 \sqrt { 65 } e ^ { t }$ , $a _ { \mathrm { N } } = \sqrt { 65 } e ^ { t }$
C) $a _ { \mathbf { T } } = 65 e ^ { t }$ , $a _ { \mathrm { N } } ( t ) = 8 e ^ { t }$
D) $a _ { \mathbf { T } } = 8 e ^ { t }$ , $a _ { \mathrm { N } } = 65 e ^ { t }$

Question

Quizplus · Accepted Answer

The scalar tangential component of acceleration is given by $$a_T = \sqrt{a_x^2+a_y^2}$$and the scalar normal component of acceleration is given by $$a_N = \frac{|a\times v|}{|v|}$$where $a$ is the acceleration vector, $v$ is the velocity vector, and $|\cdot|$ denotes the magnitude of a vector.In this case, we have$$a = \frac{d^2\mathbf{r}}{dt^2} = e^t(8\cos 8t, -64\sin 8t, 0)$$and$$v = \frac{d\mathbf{r}}{dt} = e^t(8\sin 8t, 8\cos 8t, 8)$$So,$$a_T = \sqrt{(8)^2+(-64)^2}e^t = 8\sqrt{65}e^t$$and$$a_N = \frac{|a\times v|}{|v|} = \frac{|e^t(0, 0, 640)|}{|e^t(8\sin 8t, 8\cos 8t, 8)|} = 8\sqrt{65}e^t$$

Find the Scalar Tangential and Normal Components of Acceleration of a Particle