A body of mass M thrown horizontally with velocity v from the top of the tower of height H touches the ground at a distance of $$100 \mathrm{~m}$$ from the foot of the tower. A body of mass $$2 \mathrm{~M}$$ thrown at a velocity $$\frac{v}{2}$$ from the top of the tower of height $$4 \mathrm{H}$$ will touch the ground at a distance of _______ m.

The maximum height reached by a projectile is $$64 \mathrm{~m}$$. If the initial velocity is halved, the new maximum height of the projectile is ______ $$\mathrm{m}$$.

A ball rolls off the top of a stairway with horizontal velocity $$u$$. The steps are $$0.1 \mathrm{~m}$$ high and $$0.1 \mathrm{~m}$$ wide. The minimum velocity $$u$$ with which that ball just hits the step 5 of the stairway will be $$\sqrt{x} \mathrm{~ms}^{-1}$$ where $$x=$$ __________ [use $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$$ ].

A particle starts from origin at $$t=0$$ with a velocity $$5 \hat{i} \mathrm{~m} / \mathrm{s}$$ and moves in $$x-y$$ plane under action of a force which produces a constant acceleration of $$(3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2$$. If the $$x$$-coordinate of the particle at that instant is $$84 \mathrm{~m}$$, then the speed of the particle at this time is $$\sqrt{\alpha} \mathrm{~m} / \mathrm{s}$$. The value of $$\alpha$$ is _________.