The coordinates of a moving particle at any time $t$ are given by $x = a\, t^2$ and $y = b\, t^2$. The speed of the particle is

$A$ body starts from rest from the origin with an acceleration of $6\,m/s^2$ along the $x$-axis and $8\,m/s^2$ along the $y$-axis. Its distance from the origin after $4\,s$ will be........$m$

$A$ particle moves in space according to the equation $\vec{r} = (\sin t \hat{i} + \cos t \hat{j} + t \hat{k}) \text{ m}$. Find the time $t$ when the position vector and acceleration vector are perpendicular to each other.

Two bodies are projected from points $(0,0)$ and $(\sqrt{3}-1,0)$ with velocities $10 \ ms^{-1}$ and $v \ ms^{-1}$ respectively,as shown in the figure. The time after which they collide in space is . . . . . . . (in $s$)

$A$ man standing on the roof of a house of height $h$ throws one particle vertically downwards and another particle horizontally with the same velocity $u$. The ratio of their velocities when they reach the earth's surface will be

$A$ body $P$ is projected at an angle of $30^{\circ}$ with the horizontal and another body $Q$ is projected at an angle of $30^{\circ}$ with the vertical. If the ratio of the horizontal ranges of the bodies $P$ and $Q$ is $1: 2$, then the ratio of the maximum heights reached by the bodies $P$ and $Q$ is