Find the speed of a particle with the given position function. $\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 } \mathbf { k }$
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

We need to find the speed of the particle, which is given by the magnitude of the velocity vector. Thus, we need to find the derivative of the position function to get the velocity vector and then find its magnitude.
$$\mathrm { r } ( t ) = t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 }\mathbf { k }$$
$$\mathbf { v } ( t ) = \mathrm { r } ^ { \prime } ( t ) = \frac { d } { d t } ( t \mathbf { i } + 5 t ^ { 2 } \mathbf { j } + 3 t ^ { 6 }\mathbf { k } ) = \mathbf { i } + 10 t \mathbf { j } + 18 t ^ { 5 } \mathbf { k }$$
$$|\mathbf { v } ( t )|=\sqrt{1+100t^2+324t^{10}}$$
Thus, the correct option is (D).

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