Galileo writes that for angles of projection of a projectile at angles $(45^\circ + \theta)$ and $(45^\circ - \theta)$,the horizontal ranges described by the projectile are in the ratio of (if $\theta \le 45^\circ$):

$A$ ball is thrown upwards at an angle of $60^o$ to the horizontal. It falls on the ground at a distance of $90 \,m$. If the ball is thrown with the same initial velocity at an angle $30^o$,it will fall on the ground at a distance of ........ $m$.

$A$ body is projected at an angle $\theta$ so that its range is maximum. If $T$ is the time of flight, then the value of maximum range is (acceleration due to gravity $= g$)

$A$ heavy particle is projected from a point on the horizontal at an angle $60^\circ$ with the horizontal with a speed of $10 \ m/s$. Then the radius of curvature of its path at the instant of crossing the same horizontal will be ......... $m$.

$A$ projectile is projected at $30^{\circ}$ from horizontal with initial velocity $40\,m/s$. The velocity of the projectile at $t = 2\,s$ from the start will be $........$ (Given $g = 10\,m/s^2$)

In projectile motion,the modulus of the rate of change of velocity is: