When a car is at rest,its driver sees rain drops falling on it vertically. When driving the car with speed $v$,he sees that rain drops are coming at an angle $60^{\circ}$ from the horizontal. On further increasing the speed of the car to $(1+\beta)v$,this angle changes to $45^{\circ}$. The value of $\beta$ is close to...

$A$ man of mass $60 \,kg$ is standing on a boat of mass $140 \,kg$,which is at rest in still water. The man is initially at $20 \,m$ from the shore. He starts walking on the boat for $4 \,s$ with a constant speed of $1.5 \,m/s$ towards the shore. The final distance of the man from the shore is .......... $m$.

Two girls are standing at the ends $A$ and $B$ of a ground where $AB = b$. The girl at $B$ starts running in a direction perpendicular to $AB$ with velocity $V_1$. The girl at $A$ starts running simultaneously with velocity $V_2$ and meets the other girl in the shortest possible time $t$. The value of $t$ is:

$A$ swimmer wants to cross a river from point $A$ to point $B$. Line $AB$ makes an angle of $30^{\circ}$ 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 in degrees,so that the swimmer reaches point $B$.

$A$ shell is fired at a speed of $200\ m/s$ at an angle of $37^o$ above the horizontal from the top of a tower $80\ m$ high. At the same instant,another shell is fired from a jeep travelling away from the tower at a speed of $10\ m/s$ as shown. The velocity of this shell relative to the jeep is $250\ m/s$ at an angle of $53^o$ with the horizontal. Find the time (in $sec$) taken by the two shells to come closest.

$A$ man projects a coin upwards from the gate of a uniformly moving train. The path of the coin for the man will be