A projectile is fired at an angle of $30^{\circ}$ to the horizontal such that the vertical component of its initial velocity is $80\,m / s$. Its time of flight is $T$. Its velocity at $t=\frac{T}{4}$ has a magnitude of nearly $........\frac{m}{s}$

A stone is projected in air. Its time of flight is $3\,s$ and range is $150\,m$ Maximum height reached by the stone is $......\,m$ $\left(g=10\,ms ^{-2}\right)$

Show that for a projectile the angle between the velocity and the $x$ -axis as a function of time is given by

$\theta(t)=\tan ^{-1}\left(\frac{v_{0 y}-g t}{v_{0 x}}\right)$

Show that the projection angle $\theta_{0}$ for a projectile launched from the origin is given by

$\theta_{0}=\tan ^{-1}\left(\frac{4 h_{m}}{R}\right)$

Where the symbols have their usual meaning.

From the ground level, a ball is to be shot with a certain speed. Graph shows the range $(R)$ of the particle versus the angle of projection from horizontal ( $\theta $ ). Values of $\theta _1$ and $\theta _2$ are

A particle is projected from a horizontal plane ($x-z$ plane) such that its velocity vector at time t is given by $\vec V = a\hat i + (b - ct)\hat j$ Its range on the horizontal plane is given by

A ball is projected with a velocity, $10 ms ^{-1}$, at an angle of $60^{\circ}$ with the vertical direction. Its speed at the highest point of its trajectory will be$............... ms ^{-1}$