The magnitude of acceleration and velocity of a particle moving in a plane,whose position vector is $\vec{r} = 3t^2 \hat{i} + 2t \hat{j} + \hat{k}$ at $t = 2 \text{ s}$,are,respectively:

The $x$ and $y$ coordinates of a particle at any time $t$ are given by $x = 5t - 2t^2$ and $y = 10t$ respectively,where $x$ and $y$ are in meters and $t$ is in seconds. The acceleration of the particle at $t = 2 \, s$ is . . . . . . $m/s^2$.

$A$ particle starts from the origin at $t=0 \; s$ with a velocity of $10.0 \hat{j} \; m/s$ and moves in the $x-y$ plane with a constant acceleration of $(8.0 \hat{i} + 2.0 \hat{j}) \; m/s^2$.
$(a)$ At what time is the $x$-coordinate of the particle $16 \; m$? What is the $y$-coordinate of the particle at that time?
$(b)$ What is the speed of the particle at that time?

$A$ particle initially at the origin starts moving in the $xy$-plane with a velocity component $\vec{V} = (6 + 2t) \hat{i} + (4 + 2\sqrt{3}t) \hat{j} \text{ m/s}$. The acceleration of the particle in $\text{m/s}^2$ is ($x, y$ are measured in meters,$t$ in seconds,respectively).

$A$ man moves in the $x-y$ plane along the path shown. At what point is his average velocity vector in the same direction as his instantaneous velocity vector? The man starts from point $P$.

The position of a particle is given by $r = 3.0 t \hat{i} + 2.0 t^{2} \hat{j} + 5.0 \hat{k}$,where $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres. $(a)$ Find $v(t)$ and $a(t)$ of the particle. $(b)$ Find the magnitude and direction of $v(t)$ at $t = 1.0 \ s$.