$A$ person aiming to reach the exactly opposite point on the bank of a stream is swimming with a speed of $0.5\, m/s$ at an angle of $120^\circ$ with the direction of flow of water. The speed of water in the stream is..........$m/s$.

$A$ steam boat goes across a lake and comes back: $(i)$ on a quiet day when the water is still and $(ii)$ on a rough day when there is a uniform current so as to help the journey onward and to impede the journey back. If the speed of the launch,on both days,was same,the time required for the complete journey on the rough day as compared to that on the quiet day,will be

$A$ butterfly is flying with a velocity $4 \sqrt{2} \, m/s$ in the North-East direction. The wind is blowing at $1 \, m/s$ from North to South. The resultant displacement of the butterfly in $3 \, s$ is $.... \, m$.

Ship $A$ is sailing towards north-east with velocity $\vec{v}_A = 30\hat{i} + 50\hat{j}\,\text{km/hr}$,where $\hat{i}$ points east and $\hat{j}$ points north. Ship $B$ is at a distance of $80\,\text{km}$ east and $150\,\text{km}$ north of Ship $A$ and is sailing towards west at $10\,\text{km/hr}$. After how many hours will Ship $A$ be at the minimum distance from Ship $B$?

$A$ man standing on a road has to hold his umbrella at $30^{\circ}$ with the vertical to keep the rain away. He throws the umbrella and starts running at $10 \,km/h$. He finds that raindrops are hitting his head vertically. The actual speed of raindrops 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 $B$.