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. \(\frac{g R}{\sqrt{g\cos \theta}}\)