A boat is moving with a velocity \(3\hat i + 4\hat j\) with respect to ground. The water in the river is moving with a velocity\(-3\hat i - 4 \hat j\) with respect to ground. The relative velocity of the boat with respect to water is: 
1. \(8\hat j\)
2. \(-6\hat i-8\hat j\)
3. \(6\hat i+8\hat j\)
4. \(5\sqrt{2}\)

A river is flowing from east to west at a speed of \(5\) m/min. A man on south bank of river, capable of swimming \(10\) m/min in still water, wants to swim across the river in shortest time. He should swim:

A person, aiming to reach the exact opposite point on the other bank of a stream, is swimming with a speed of \(0.5\ \text{m/s}\) at an angle of \(120^\circ\) to the direction of the flow of water. The speed of water in the stream is:

1. \(1\ \text{m/s}\)

2. \(0.5\ \text{m/s}\)

3. \(0.25\ \text{m/s}\)

4. \(0.433\ \text{m/s}\)

A thief is running away on a straight road in a jeep moving with a speed of \(9\ \text{m/s}\). A police man chases him on a motorcycle moving at a speed of \(10\ \text{m/s}\). If the instantaneous separation of the jeep from the motorcycle is \(100\ \text{m}\), how long will it take for the policeman to catch the thief:

1. \(1\) second

2. \(19\) second

3. \(90\) second

4. \(100\) second

A man sitting in a bus travelling in a direction from west to east with a speed of \(40\) km/h observes that the rain-drops are falling vertically downwards. To another man standing on ground the rain will appear: 

A boat takes two hours to travel \(8\ \text{km}\) and back in still water. If the velocity of water is \(4\ \text{km/h}\), the time taken for going upstream \(8\ \text{km}\) and coming back is:

1. \(2\ \text{h}\)

2. \(2\ \text{h}\ 40\ \text{min}\)

3. \(1\ \text{h}\ 20\ \text{min}\)

4. Cannot be estimated with the information given

A \(120\ \text{m}\) long train is moving towards west with a speed of \(10\ \text{m/s}\). A bird flying towards east with a speed of \(5\ \text{m/s}\) crosses the train. The time taken by the bird to cross the train will be:

1. \(16\ \text{s}\)

2. \(12\ \text{s}\)

3.  \(10\ \text{s}\)

4. \(8\ \text{s}\)

A boat crosses a river with a velocity of \(8\ \text{km/h}\). If the resulting velocity of the boat is \(10\ \text{km/h}\), then the velocity of the river water is:

1. \(4\ \text{km/h}\)

2. \(6\ \text{km/h}\)

3. \(8\ \text{km/h}\)

4. \(10\ \text{km/h}\)

A vector \(\vec{a}\) is turned without a change in its length through a small angle \(d\theta\). The values of \(\left | \Delta \vec{a} \right |\) and \(\Delta a\) are respectively:
1. $\mathrm{}$\(0,\)  \(ad\theta\)
2. \(ad\theta,\) \(0\)
3. \(0,\) \(0\)
4. none of these

A steam boat goes across a lake and comes back (a) On a quiet day when the water is still and (b) On a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both the days was the same, in which case will the steam boat complete the journey in lesser time: