Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }$
A) $| v ( t ) | = 5 \left( e ^ { 5 t } + e ^ { - 5 t } \right)$
B) $| v ( t ) | = \left( e ^ { 5 t } + e ^ { - 5 t } \right)$
C) $| v ( t ) | = \sqrt { 5 + 5 e ^ { t } + 5 e ^ { - t } }$
D) $| v ( t ) | = 5 \left( e ^ { t } + e ^ { - t } \right)$
E) $| v ( t ) | = \sqrt { 5 + e ^ { 5 t } + e ^ { - 50 t } }$

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: $\mathbf{v}(t) = \frac{d}{dt}(5\sqrt{2}t \mathbf{i} + e^{5t} \mathbf{j} - e^{-5t} \mathbf{k}) = 5\sqrt{2} \mathbf{i} + 5e^{5t} \mathbf{j} + 5e^{-5t} \mathbf{k}$. The magnitude of this velocity vector is $|v(t)| = \sqrt{(5\sqrt{2})^2 + (5e^{5t})^2 + (5e^{-5t})^2} = 5\sqrt{2 + e^{10t} + e^{-10t}}$. Simplifying gives $|v(t)| = 5(e^{5t} + e^{-5t})$.

Find the Speed of a Particle with the Given Position r(t)=52ti+e5tj−e−5tk\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }r(t)=52​ti+e5tj−e−5tk

Find the Speed of a Particle with the Given Position $\mathbf { r } ( t ) = 5 \sqrt { 2 } t \mathbf { i } + e ^ { 5 t } \mathbf { j } - e ^ { - 5 t } \mathbf { k }$