A swimmer can swim in still water at a speed of \(5~\text{m/s}.\) Let \(t_1~\text{and}~t_2,\)​ be the times taken to cross a river of width \(d\) by the shortest-time path and by the shortest path, respectively. If the speed of the river current increases, the times to cross the river along the shortest-time path and shortest path become \(t'_1~\text{and}~t'_2,\) respectively. Which of the following statements is correct?
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) $\frac{\mathrm{R}}{2}<mspace\; width="1em"></mspace>\cot <mspace\; width="1em"></mspace>\mathrm{\theta }$

(2) $\frac{\mathrm{R}}{2}<mspace\; width="1em"></mspace>\tan <mspace\; width="1em"></mspace>\mathrm{\theta }$

(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:

$\mathrm{x}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{\mathrm{B}}{\mathrm{t}}$, $\mathrm{y}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\mathrm{A}\sqrt{\mathrm{t}}$.

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.  $\frac{\mathrm{u}<mspace\; width="1em"></mspace>{\sin }^2<mspace\; width="1em"></mspace>\mathrm{\theta }}{\mathrm{g}}$

Six friends stand at the six vertices of a regular hexagonal park. Each friend always maintains a direction towards the friend at the next corner with a constant speed of \(2~\text{m/s}.\) They eventually meet after \(60~\text{s}.\) What is the length of each side of the hexagon?
1. \(120~\text{m}\)
2. \(30~\text{m}\)
3. \(240~\text{m}\)
4. \(60~\text{m}\)

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 ${\mathrm{t}}_1<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>{\mathrm{t}}_2$ respectively (taking air friction into consideration), then

(1) ${\mathrm{t}}_1<mspace\; width="1em"></mspace>><mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(2) ${\mathrm{t}}_1<mspace\; width="1em"></mspace><<mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(3) ${\mathrm{t}}_1<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>{\mathrm{t}}_2$

(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) $\mathrm{t}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{\mathrm{u}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\mathrm{\theta }}{\mathrm{g}}$

(2) $\mathrm{t}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{\mathrm{u}}{\mathrm{g}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\mathrm{\theta }}$

(3) $\mathrm{t}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{\mathrm{u}}{\mathrm{g}<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>\mathrm{\theta }}$

(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 \(10/\pi^2]\)