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 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 particle moves towards the east with a velocity \(5\ \text{m/s}\). After \(10\) seconds its direction changes towards the north with the same velocity. The average acceleration of the particle is:

1. Zero

2. \(\dfrac{1}{\sqrt{2}}\,m/s^2\ \text{N-W}\)

3. \(\dfrac{1}{\sqrt{2}}\,m/s^2\ \text{N-E}\)

4. \(\dfrac{1}{\sqrt{2}}\,m/s^2\ \text{S-W}\)

To a person going eastward in a car with a velocity of \(25\ \text{ km/hr}\), a train appears to move towards the north with a velocity of \(25\sqrt3\ \text{ km/hr}\). The actual velocity of the train will be:

1. \(25\ \text{ km/hr}\)

2. \(50\ \text{ km/hr}\)

3. \(5\ \text{ km/hr}\)

4.  \(5\sqrt3\ \text{ km/hr}\)$\mathrm{}$

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 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 man standing on a road holds his umbrella at 30° with the vertical to keep the rain away. He throws the umbrella and starts running at 10 km/hr. He finds that raindrops are hitting his head vertically, the speed of raindrops with respect to the road will be:

1. 10 km/hr

2. 20 km/hr

3. 30 km/hr

4. 40 km/hr

The \(x\) and \(y\) coordinates of the particle at any time are \(x = 5t-2t^2\) and \(y=10t\) respectively, where \(x\) and \(y\) are in metres and \(t\) is in seconds. The acceleration of the particle at \(t=2\) s is:
1. \(0\) m/s²
2. \(5\) m/s²
3. \(-4\) m/s²
4. \(-8\) m/s²