A projectile is launched from the origin in the $xy$ plane ( $x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance. $r =\sqrt{ x ^2+ y ^2}$ from the origin is plotted against $x$, the resulting curves show different behaviours for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure below. For $\alpha_1, r ( x )$ keeps increasing with $x$ while for $\alpha_2$, $r(x)$ increases and reaches a maximum, then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at an angle $\alpha_c\left(\alpha_1 < \alpha_c < \alpha_2\right)$. The value of $\alpha_c$ is [ignore where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity]
$\sin ^{-1}\left(\frac{1}{3}\right)$
$\cos ^{-1}\left(\frac{1}{3}\right)$
$\tan ^{-1}\left(\frac{1}{3}\right)$
$\tan ^{-1}(3)$
A projectile projected at an angle ${30^o}$ from the horizontal has a range $2\upsilon ,\,\sqrt 2 \upsilon \,\,{\rm{ and}}\,{\rm{zero}}$. If the angle of projection at the same initial velocity be ${60^o}$, then the range will be
A projectile crosses two walls of equal height $H$ symmetrically as shown If the horizontal distance between the two walls is $d = 120\,\, m$, then the range of the projectile is ........ $m$
A wheel of radius $R$ is trapped in a mud pit and spinning. As the wheel is spinning, it splashes mud blobs with initial speed $u$ from various points on its circumference. The maximum height from the centre of the wheel, to which a mud blob can reach is
If we can throw a ball upto a maximum height $H$, the maximum horizontal distance to which we can throw it is
A projectile is fired at $30^{\circ}$ to the horizontal, The vertical component of its velocity is $80 \;ms ^{-1}$, Its time flight is $T$. What will be the velocity of projectile at $t =\frac{ T }{2}$?