Find the speed of a particle with the given position function. $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$
A) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 100 t ^ { 10 } }$
B) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t + 324 t ^ { 9 } }$
C) $| \mathbf { v } ( t ) | = 1 + 100 t ^ { 2 } + 324 t ^ { 10 }$
D) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t ^ { 2 } + 324 t ^ { 10 } }$
E) $| \mathbf { v } ( t ) | = \sqrt { 1 + 100 t + t ^ { 9 } }$

Question

Quizplus · Accepted Answer

To find the speed of the particle, we first need to find its velocity function:

\begin{align*}
\mathbf{v}(t) &= \frac{d}{dt} r(t) \
&= \frac{d}{dt} (t\mathbf{i} + 5t^2\mathbf{j} + 3t^6\mathbf{k}) \
&= \mathbf{i} + 10t\mathbf{j} + 18t^5\mathbf{k}
\end{align*}

Then, the speed at any time t is given by the magnitude of the velocity:

$$|\mathbf{v}(t)| = \sqrt{1 + 100t^2 + 324t^{10}}$$

Therefore, the correct choice is D.

Find the Speed of a Particle with the Given Position r(t)=ti+5t2j+3t6kr ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }r(t)=ti+5t2j+3t6k

Find the Speed of a Particle with the Given Position $r ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$