A particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t )$, and the acceleration of the particle at time $t$ is given by $a ( t )$. Which of the following gives the average velocity of the particle from time $t = 0$ to time $t = 8$ ?
(A) $\frac { a ( 8 ) - a ( 0 ) } { 8 }$
(B) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(C) $\frac { 1 } { 8 } \int _ { 0 } ^ { 8 } | v ( t ) | d t$
(D) $\frac { 1 } { 2 } \int _ { 0 } ^ { 8 } v ( t ) d t$
(E) $\frac { v ( 0 ) + v ( 8 ) } { 2 }$

A particle moves along a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

45-- The position--time equation of a particle in SI units is $x = 2t^2 - 12t + 8$, at moment $t = 0$. How many seconds after $t = 0$ is the particle's distance from the origin less than or equal to 8 meters?
(1) $2$ (2) $3$ (3) $4$ (4) $6$
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$\vec{\mathrm{F}} = 4\mathrm{t}^{3}\hat{\mathrm{i}} - 3\mathrm{t}^{2}\hat{\mathrm{j}}, \mathrm{m} = 4\mathrm{~kg}$ at $\mathrm{t} = 0$ particle is at rest and at origin then find velocity and position at $\mathbf{t} = \mathbf{2}\mathbf{~sec}$.