A projectile is given an initial velocity of \(\hat i+2\hat j\). The cartesian equation of its path is: (Take \(g = 10 \ \text{ms}^{ - 2}\)) 

1. \(y=2x-5x^2\)

2. \(y=x-5x^2\) $\mathrm{}$

3. \(4y=2x-5x^2\)$\mathrm{}$

4. \(y=2x-25x^2\)

Ship A is moving westward with a speed of \(10\ \text{km h}^{- 1}\) and Ship B, south of A, is moving northwards with a speed of \(10\ \text{km h}^{- 1}\)${\mathrm{}}^{}$. The time after which the distance between them becomes the shortest is:

1. \(5\) hr

2. \(5 \sqrt2\) hr

3. \(10 \sqrt2\)$\mathrm{}$ hr

4. \(0\) hr

The time taken by the projectile to reach from \(A\) to \(B\) is \(t\). Then the distance \(AB\) is equal to :

1. \(\dfrac{ut}{\sqrt3}\)

2. \(\dfrac{\sqrt3ut}{2}\)

3. \(\sqrt3ut\)

4. \(2ut\)

A particle projected with kinetic energy ${\mathrm{k}}_0$ with an angle of projection $\mathrm{\theta }$. Then the variation of kinetic K with vertical displacement y is

1.  linear

2.  parabolic

3.  hyperbolic

4.  periodic

A river is flowing with a speed of \(1\ \text{km/hr}\). A swimmer wants to go to point \(C\) starting from \(A\). He swims with a speed of \(5\ \text{km/hr}\), at an angle \(\theta\)  with respect to the river. If \(\text {AB = BC} = 400\ \text{m}\), then:

1. The time taken by the man is \(12\) min

2. The time taken by the man is \(8\) min

3. The value of \(\theta\) is \(45^\circ\)

4. The value of \(\theta\) is \(53^\circ\)

 A body is thrown horizontally with a velocity \(\sqrt{2 g h}\) from the top of a tower of height \(h\). It strikes the level ground through the foot of the tower at a distance \(x\) from the tower. The value of \(x\) is:

Two men \(P\) and \(Q\) are standing at corners \(A\) and \(B\) of a square \(ABCD\) of side \(8~\text m.\) They start moving along the track with a constant speed \(2~\text{m/s}\) and \(10~\text {m/s}\) respectively. The time when they will meet for the first time is equal to:
       
1. \(2~\text{sec}\)
2. \(3~\text{sec}\)
3. \(1~\text{sec}\)
4. \(6~\text{sec}\) 

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 (at t=0) is:

1. variable

2. $\sqrt{\frac{2a}{b}}$

3. $\frac{a}{2b}$

4. $\sqrt{\frac{a}{2b}}$

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