An ant is moving on a plane horizontal surface. The number of degrees of freedom of the ant will be .........

Two particles $A$ and $B$ are moving in the $XY$-plane. Their positions vary with time $t$ according to the relations:
$x_A(t) = 3t, \quad x_B(t) = 6$
$y_A(t) = t, \quad y_B(t) = 2 + 3t^2$
What is the distance between the two particles at $t = 1$?

$A$ particle moves in the $x-y$ plane with velocity $\vec{v} = a\hat{i} + bx\hat{j}$,where $a$ and $b$ are constants. If the particle was initially at the origin,find the equation of its trajectory.

The $x$ and $y$ coordinates of a particle at any time $t$ are given by $x = 7t + 4t^2$ and $y = 5t$,where $x$ and $y$ are in $m$ and $t$ is in $s$. The acceleration of the particle at $t = 5 \ s$ is ......... $m/s^2$.

Position of an ant ($S$ in metres) moving in $Y-Z$ plane is given by $S = 2t^2 \hat{j} + 5 \hat{k}$ (where $t$ is in seconds). The magnitude and direction of velocity of the ant at $t = 1 \ s$ will be:

Derive the equations of motion for a body moving in two dimensions: $\vec{v} = \vec{v_0} + \vec{a}t$ and $\vec{r} = \vec{r_0} + \vec{v_0}t + \frac{1}{2}\vec{a}t^2$.