$A$ body is projected from the top of a tower with a velocity $\overrightarrow{u} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \text{ m/s}$, where $\hat{i}, \hat{j},$ and $\hat{k}$ are unit vectors along east, north, and vertically upwards, respectively. If the height of the tower is $30 \text{ m}$, find the horizontal range of the body on the ground. (Take $g = 10 \text{ m/s}^2$) (in $\text{ m}$)

$A$ gun can fire shells with maximum speed $v_0$ and the maximum horizontal range that can be achieved is $R = \frac{v_0^2}{g}$. If a target farther away by distance $\Delta x$ (beyond $R$) has to be hit with the same gun,show that it could be achieved by raising the gun to a height at least $h = \Delta x \left[ 1 + \frac{\Delta x}{R} \right]$.

$A$ ball is projected upwards from the top of a tower with a velocity $50 \, m/s$ making an angle $30^\circ$ with the horizontal. The height of the tower is $70 \, m$. After how many seconds from the instant of throwing will the ball reach the ground? ........ $s$

Two particles in the same vertical plane are thrown to strike at the same time. One is thrown from the ground and the other from a height $h$ vertically above it. The ground particle is thrown obliquely with speed $u$ at an angle $\theta$ and achieves a maximum height $H$. The second particle is thrown horizontally with the same speed $u$. What is the maximum height $h$ such that the two particles strike each other in the air?

At a height $0.4\, m$ from the ground,the velocity of a projectile in vector form is $\vec v = (6\hat i + 2\hat j)\,m/s$. The angle of projection is ...... $^o$ $(g = 10\, m/s^2)$.

$A$ child stands on the edge of a cliff $10 \ m$ above the ground and throws a stone horizontally with an initial speed of $5 \ m/s$. Neglecting air resistance,the speed with which the stone hits the ground will be $.......... \ m/s$ (given,$g = 10 \ m/s^2$).