Find the speed of a particle with the given position function.
$$\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$$
A) $| v ( t ) | = \sqrt { 4 + 4 e ^ { t } + 4 e ^ { - t } }$
B) $| v ( t ) | = \left( e ^ { 4 t } + e ^ { - 4 t } \right)$
C) $| v ( t ) | = \sqrt { 4 + e ^ { 4 t } + e ^ { - 40 t } }$
D) $| v ( t ) | = 4 \left( e ^ { 4 t } + e ^ { - 4 t } \right)$
E) $| v ( t ) | = 4 \left( e ^ { t } + e ^ { - t } \right)$

Question

Quizplus · Accepted Answer

The speed of a particle is the magnitude of its velocity vector. First, find the velocity vector by differentiating the position function: $\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = 4\sqrt{2} \mathbf{i} + 4e^{4t} \mathbf{j} + 4e^{-4t} \mathbf{k}$. The magnitude of this vector is $|v(t)| = \sqrt{(4\sqrt{2})^2 + (4e^{4t})^2 + (4e^{-4t})^2} = 4\sqrt{2 + e^{8t} + e^{-8t}}$. Simplifying gives the correct choice.

Find the Speed of a Particle with the Given Position r(t)=42ti+e4tj−e−4tk\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }r(t)=42​ti+e4tj−e−4tk

Find the Speed of a Particle with the Given Position $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$