Find the average velocity of a projectile between the instants it crosses half the maximum height. (It is projected with a speed $u$ at an angle $\theta$ with the horizontal.)

A staircase contains three steps, each $10 \mathrm{cm}$ high and $20 \mathrm{cm}$ wide. What should be the minimum horizontal velocity of a ball rolling off the uppermost plane so as to hit the lowest plane directly?

A person is standing on a truck moving with a constant velocity of $14.7 \mathrm{m} {\mathrm{s}}^{-1}$ on a horizontal road. The man throws a ball in such a way that it returns to the truck after the truck has moved $58.8 \mathrm{m}$. Find the speed and the angle of projection,

(a) as seen from the truck.

(b) as seen from the ground.

A river $400 \mathrm{m}$ wide is flowing at a rate of $2.0 \mathrm{m} {\mathrm{s}}^{-1}$. A boat is sailing at a velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ with respect to the water in a direction perpendicular to the river.

(a) Find the time taken by the boat to reach the opposite bank.

(b) How far from the point directly opposite to the starting point does the boat reach the opposite bank?

A swimmer wishes to cross a $500 \mathrm{m}$ wide river flowing at $5 \mathrm{km} {\mathrm{h}}^{-1}$. His speed with respect to water is $3 \mathrm{km} {\mathrm{h}}^{-1}$. 

(a) If he heads in a direction making an angle $\text{θ}$ with the flow, find the time he takes to cross the river.

(b) Find the shortest possible time to cross the river.