A particle moves in $X-Y$ plane with constant acceleration $a$ and is directed along the negative $Y$-axis. The equation of motion of the particle has the form $Y=px-q{x}^2,$ where $p$ and $q$? are positive constants. The velocity of the particle at the origin is

Two projectiles $A$ and $B$ are thrown from the same point with velocities $2v$ and $v$ respectively. If $B$ was thrown at an angle $45°$ with the horizontal, find the angle with which $A$ was projected, if it is known that both $A$ and $B$ have equal ranges ?

The time period of a particle $P$ moving along a circular path of radius $1 \mathrm{cm}$ is $\pi s.$  Another particle $Q$ starts from rest at a point $A$ and travelling along a straight line path reaches the point $B$ in $2 \mathrm{s}$. If $AB=12 \mathrm{cm}$ and the particles $P$ and $Q$ are respectively at $D$ and $B$ at an instant shown in the figure, the acceleration of particle $Q$ with respect to particle $P$ is _______$\mathrm{cm} {\mathrm{s}}^{-2}$. (Points $A\mathit{,}B\mathit{,}C\mathit{,}D$ and $O$ lie along the same straight line)