A racing car moves along a circular track of radius $b$. The car starts from rest and its speed increases at a constant rate $\alpha$. Let the angle between the velocity and the acceleration be $\theta$ at time $t$. Then, $\cos \theta$ is,

Two identical discs of the same radius $R$ are rotating about their axes in opposite directions with the same constant angular speed $\omega$. The disc is in the same horizontal plane.At time $t\mathit{ }= 0$, the points $P$ and $Q$ are facing each other as shown in the figure. The relative speed between the two points $P$ and $Q$ is $v_{\mathit{r}}$ as a function of times best represented by

 Two men, $P$ and $Q$ are standing at corners $A$ and $B$ of a square $ABCD$ of side $8 \mathrm{m}$. They start moving along the track with constant speed $2 \mathrm{m} {\mathrm{s}}^{-1}$ and $10 \mathrm{m} {\mathrm{s}}^{-1}$, respectively. Find the time when they will meet for the first time.

$P$ is a point moving with a constant speed $10 \mathrm{m} {\mathrm{s}}^{-1}$ such that its velocity vector always maintains an angle $60°$ with the line $OP$, as shown in the figure (where $O$ is a fixed point in space). The initial distance between $O$ and $P$ is $100 \mathrm{m}$. After what time (in $\sec$) shall $P$ reach $O$? 

A train of length $210 \mathrm{m}$ is moving with a speed of $25 \mathrm{m} {\mathrm{s}}^{-1}$ in North direction. A small bird is flying above the train with a speed of $5 \mathrm{m} {\mathrm{s}}^{-1}$ in south direction. Find the time taken by the bird to cross the train in seconds.