A swimmer can swim in still water at a speed of \(5~\text{m/s}\) and takes \(t_1~\text{and}~t_2,\) time to cross a river in the shortest time and shortest path respectively. If the speed of the river increases, the time is taken to cross the river in the shortest time and the shortest path be \(t'_1~\text{and}~t'_2\) respectively. Then:
1. \(t_1<t'_1, t_2< t'_2\)
2. \(t_1>t'_1, t_2> t'_2\)
3. \(t_1=t'_1, t_2< t'_2\)
4. \(t_1=t'_1, t_2> t'_2\)

A sprinkler is deployed to irrigate the garden. The speed of water-jet from the sprinkler is u. The maximum area which can be irrigated by the sprinkler is

1.  $\frac{{\mathrm{\pi }}^2{\mathrm{u}}^2}{\mathrm{g}}$

2.  $\frac{{\mathrm{\pi u}}^2}{\mathrm{g}}$

3.  $\frac{{\mathrm{\pi u}}^2}{{\mathrm{g}}^2}$

4.  $\frac{{\mathrm{\pi u}}^4}{{\mathrm{g}}^2}$

Which of the following is an appropriate expression for the radius of curvature of a projectile at the highest point? (R $\to$ Range of projectile)

(1) $\frac{\mathrm{R}}{2} \cot \mathrm{\theta }$

(2) $\frac{\mathrm{R}}{2} \tan \mathrm{\theta }$

(3) 2R cot $\mathrm{\theta }$

(4) 2R tan $\mathrm{\theta }$

The coordinates of a particle moving in the x-y plane vary with time according to the following relation:

$\mathrm{x} = \frac{\mathrm{B}}{\mathrm{t}}$, $\mathrm{y} = \mathrm{A}\sqrt{\mathrm{t}}$.

The locus of the particle is:

(1) Parabola

(2) Circle

(3) Eclipse

(4) Hyperbola

If ${\mathrm{t}}_1$ represents the instant at which the instantaneous velocity becomes perpendicular to the direction of initial velocity during a projectile and ${\mathrm{t}}_2$ represents the time after which the particle attains maximum height, then $\sqrt{{\mathrm{t}}_1{\mathrm{t}}_2}$ is:

1.  $\frac{\mathrm{u}}{\mathrm{g}}$

2.  $\frac{2\mathrm{u}}{\mathrm{g}}$

3.  $\frac{\mathrm{u}}{2\mathrm{g}}$

4.  $\frac{\mathrm{u} {\sin }^2 \mathrm{\theta }}{\mathrm{g}}$

A park is in the shape of a regular hexagon. Six friends standing at each corner of the park start moving towards each other with the same speed 2 m/s and meet each other after 60 s. The side of the park is

(1) 120 m

(2) 30 m

(3) 240 m

(4) 60 m

A ball is projected horizontally from a cliff 40 m high at a speed of 40 m/s and simultaneously a ball is dropped. If the time taken by the two balls to reach the ground are ${\mathrm{t}}_1 \mathrm{and} {\mathrm{t}}_2$ respectively (taking air friction into consideration), then

(1) ${\mathrm{t}}_1 > {\mathrm{t}}_2$

(2) ${\mathrm{t}}_1 < {\mathrm{t}}_2$

(3) ${\mathrm{t}}_1 = {\mathrm{t}}_2$

(4) Depending on air friction ${\mathrm{t}}_1$ maybe less or more than ${\mathrm{t}}_2$

 Select the incorrect statement:

A projectile is fired horizontally from the top of a tower. The time after which the instantaneous velocity will be perpendicular to the initial velocity is (neglect air resistance) :

(1) $\mathrm{t} = \frac{\mathrm{u} \sin \mathrm{\theta }}{\mathrm{g}}$

(2) $\mathrm{t} = \frac{\mathrm{u}}{\mathrm{g} \sin \mathrm{\theta }}$

(3) $\mathrm{t} = \frac{\mathrm{u}}{\mathrm{g} \cos \mathrm{\theta }}$

(4) It will never be perpendicular at any instant

A particle of mass \(2\) kg is moving in a circular path with a constant speed of \(10\) m/s. The change in the magnitude of velocity when a particle travels from \(P\) to \(Q\) will be: [assume the radius of the circle is \(10/\pi^2]\)