Two particles are projected from the two towers simultaneously, as shown in the figure. What should be value of '$d$' for their collision in meter.

A particle is projected from a point $O$ with an initial speed of $30 \mathrm{m} {\mathrm{s}}^{-1}$ to pass through a point which is $40 \mathrm{m}$ from O horizontally and $10 \mathrm{m}$ above $O$. There are two angles of projection for which this is possible. There angles are $\alpha$ and $\beta$. The value of $\left|\tan \left(\alpha +\beta \right)\right|$ is

The position-time $\left(x-t\right)$ graphs for two children $A$ and $B$ returning from their school $O$ to their home $P$ and $Q$ respectively are shown in diagram. Choose correct entries in the brackets below:

(i) $\left(\frac{A}{B}\right)$ lives closer to the school than $\left(\frac{B}{A}\right)$

(ii) $\left(\frac{A}{B}\right)$ starts from the school earlier than $\left(\frac{B}{A}\right)$

(iii) $\left(\frac{A}{B}\right)$ walks faster than $\left(\frac{B}{A}\right)$

(iv) $A$ and $B$ reach home at the (same/different) time

(v) $\left(\frac{A}{B}\right)$ overtakes $\left(\frac{B}{A}\right)$ overtakes $\left(\frac{B}{A}\right)$ on the road (once/twice).

A particle is projected horizontally as shown from the rim of a large hemispherical bowl. The displacement of the particle when it strikes the bowl the first time is $R$. Find the velocity of the particle at that instant and the time taken.