Answer:
The particle's velocity is the derivative of the particle's position. The particles's acceleration is the derivative of the particle's velocity. You can compute the velocity and acceleration as follows:
Step-by-step explanation:
[tex]\vec{v}(t)=\frac{dr}{dt}=\frac{d}{dt}(8t+5)\overrightarrow{i}+\frac{d}{dt}(8t^2-2)\overrightarrow{j}+\frac{d}{dt}(6t)\overrightarrow{k}=8\overrightarrow{i}+16t\overrightarrow{j}+6\overrightarrow{k}.[/tex]
[tex]\vec{a}(t)=\frac{dv}{dt}=\frac{d}{dt}(8)\overrightarrow{i}+\frac{d}{dt}(16t)\overrightarrow{j}+\frac{d}{dt}(6)\overrightarrow{k}=0\overrightarrow{i}+16\overrightarrow{j}+0\overrrighatarrow{k}.[/tex]
The velocity at [tex]t=0[/tex] is [tex]\vec{v}(0)=8\overrightarrow{i}+0\overrightarrow{j}+6\overrighatarrow{k}[/tex].
The speed at t=0 is [tex]\lVert \vec{v}(0)\rVert =10[/tex]. Then, the velocity at t=0 written as a product of the speed at t=0 and the direction at t=0 is
[tex]\vec{v}(0)=\lVert \vec{v}(0)\rVert \dfrac{\vec{v}(0)}{\lVert \vec{v}(0)\rVert}=10\cdot(\dfrac{8}{10},0,\dfrac{6}{10}).[/tex]
