A particle $A$ is projected vertically upwards. Another identical particle $B$ is projected at an angle of $45^o $ . Both reach the same height. The ratio of the initial kinetic energy of $A$ to that of $B$ is
$1:2$
$2:1$
$1:\sqrt 2$
$\sqrt 2:1$
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.
In projectile motion, the modulus of rate of change of velocity
A bullet fired at an angle of $30^o$ with the horizontal hits the ground $3.0\; km$ away. By adjusting its angle of projection, can one hope to hit a target $5.0\; km$ away ? Assume the muzzle speed to be fixed, and neglect air resistance.
A projectile is fired with a velocity at right angle to the slope which is inclined at an angle $\theta$ with the horizontal. The expression for the range $R$ along the incline is
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 $........\frac{m}{s}$