A particle starts from the origin at \(t=0\) and moves in the \(x-y\) plane with constant acceleration \(a\) in the \(y\) direction. Its equation of motion is \(  𝑦 = 𝑏 𝑥^2\). The \(x\) component of its velocity \((\text{at} \ t=0) \) is:

1. Variable

2. \(\sqrt{\dfrac{2a}{b}}\)

3. \(\dfrac{a}{2b}\)

4. \(\sqrt{\dfrac{a}{2b}}\)

A body is projected with a velocity \(u\) with an angle of projection \(\theta.\) The change in velocity after the time \((t)\) from the time of projection will be:

What determines the nature of the path followed by the particle?

1. Speed only

2. Velocity only

3. Acceleration only

4. None of these

A boat is sent across a river in a perpendicular direction with a velocity of \(8\ \text{km/hr}\). If the resultant velocity of the boat is \(10\ \text{km/hr}\), then the velocity of the river is: 

1. \(10\ \text{km/hr}\)

2. \(8\ \text{km/hr}\) 

3. \(6\ \text{km/hr}\) 

4. \(4\ \text{km/hr}\) 

A boat is moving with a velocity of \(3\hat i+4 \hat j\) in the river, and the water is moving with a velocity of \(-3\hat i-4 \hat j\) with respect to the ground. The relative velocity of the boat with respect to the water is: 

1. \(-6\hat i-8 \hat j\)$$

2. \(6\hat i+8 \hat j\)$\mathrm{}$

3. \(8 \hat j\)$\mathrm{}$

4. \(6 \hat j\)$\mathrm{}$

A boat moves with a speed of \(5\ \text{km/h}\) relative to water in a river flowing with a speed of \(3\ \text{km/h}\) and having a width of \(1\ \text{km}\). The minimum time taken for a round trip (returning to the initial point) is:

1. \(5\ \text{min}\)

2. \(60\ \text{min}\)

3. \(20\ \text{min}\)

4. \(30\ \text{min}\)

A river is flowing from W to E with a speed of 5 m/min. A man can swim in still water with a velocity 10 m/min. In which direction should the man swim so as to take the shortest possible path to go to the south. 

1. 30° with downstream

2. 60° with downstream

3. 120° with downstream

4. South

A train is moving towards the east and a car is along the north at the same speed. The observed direction of the car to the passenger on the train is:

A ball \(P\) is dropped vertically and another ball \(Q\) is thrown horizontally from the same height and at the same time. If air resistance is neglected, then:

1. Ball \(P\) reaches the ground first

2. Ball \(Q\) reaches the ground first

3. Both reach the ground at the same time

4. The respective masses of the two balls will decide the time

A frictionless wire \(AB\) is fixed on a sphere of radius \(R\). A very small spherical ball slips on this wire. The time taken by this ball to slip from \(A\) to \(B\) is:
          
1. \(\frac{2 \sqrt{g R}}{g \cos \theta}\)
2. \(2 \sqrt{g R} . \frac{\cos \theta}{g}\)
3. \(2 \sqrt{\frac{R}{g}}\)
4. \(\frac{g R}{\sqrt{g\cos \theta}}\)