A river has a steady speed of $v$. A man swims upstream at a distance of $d$ and swims back to the starting point in total time $t$. The man can swim at a speed of $2v$ in still water. If the time taken by the man in still water is $t_0$ to complete the same length of swim, then $\frac{t}{t_0}$ is

A swimmer wants to cross a river from point $A$ to point $B$. Line AB makes an angle of $30°$ with the flow of the river. The magnitude of the velocity of the swimmer is the same as that of the river. The angle $\theta$ with the line $AB$ should be _______$°$, so that the swimmer reaches point $B$.

The time period of a particle $P$ moving along a circular path of radius $1 \mathrm{cm}$ is $\pi s.$  Another particle $Q$ starts from rest at a point $A$ and travelling along a straight line path reaches the point $B$ in $2 \mathrm{s}$. If $AB=12 \mathrm{cm}$ and the particles $P$ and $Q$ are respectively at $D$ and $B$ at an instant shown in the figure, the acceleration of particle $Q$ with respect to particle $P$ is _______$\mathrm{cm} {\mathrm{s}}^{-2}$. (Points $A\mathit{,}B\mathit{,}C\mathit{,}D$ and $O$ lie along the same straight line)