The trajectory of a projectile in a vertical plane is $y=\alpha x-\beta x^2,$ where $\alpha$ and $\beta$ are constants and $x \& y$ are respectively the horizontal and vertical distances of the projectile from the point of projection. The angle of projection $\theta$ and the maximum height attained $H$ are respectively given by

The speed verses time graph for a particle is shown in the figure. The distance travelled (in $\mathrm{m}$) by the particle during the time interval $\mathrm{t}=0$ to $\mathrm{t}=5$ $\mathrm{s}$ will be __________.

A balloon is moving up in air vertically above a point $A$ on the ground. When it is a height $h_1,$ a girl standing at a distance d (point B) from A (see figure) sees it at an angle $45°$ with respect to the vertical. When the balloon climbs up a further height $h_2$, it is seen at an angle $60°$ with respect to the vertical if the girl moves further by a distance $2.464 \mathrm{d}$ (point C). Then the height ${\mathrm{h}}_2$ is (given $\tan 30°=0.5774$):

The velocity $\left(v\right)$ and time $\left(t\right)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at $4.333$ seconds. The total distance covered by the body in $6 \mathrm{s}$ is :