A projectile is fired from level ground at an angle $\mathrm{\theta }$ above the horizontal. The elevation angle $\varphi$ of the highest point as seen from the launch point is related to $\mathrm{\theta }$ by the relation

A particle is projected with a certain velocity at an angle above the horizontal from the foot of an inclined plane of inclination $30°$. If the particle strikes the plane normally, then $\alpha$ is equal to

In the figure, the time taken by the projectile to reach from $A$ to $B$ is $t$. Then the distance $AB$ is equal to
  

A motor cyclist is trying to jump across a path as shown in figure by driving horizontally off a cliff $A$ at a speed of $5 \mathrm{m} {\mathrm{s}}^{-1}$. Ignore air resistance and take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$. The speed with which he touches peak $B$ is