A projectile is thrown with an initial velocity (${\mathrm{v}}_{\mathrm{x}}\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>{\mathrm{v}}_{\mathrm{y}}\hat{\mathrm{j}}$)m/s. Consider horizontal direction and vertical direction as x and y axes respectively. If the range of the projectile is double the maximum height, then $\frac{{\mathrm{v}}_{\mathrm{y}}}{{\mathrm{v}}_{\mathrm{x}}}$ is

(1) 4

(2) 3

(3) 2

(4) 1

A car is moving on a circular track at a speed of 20 m/s having a radius of 100 m. If it starts decelerating uniformly at the rate of $3<mspace\; width="1em"></mspace>\mathrm{m}/{\mathrm{s}}^2$, the angle made by the resultant acceleration with the direction of its instantaneous velocity is:

(1) 37°

(2) 127°

(3) 53°

(4) 143°

An object is moving on a circular path of the radius r with constant speed v. The average acceleration of the object, after it has travelled three and a half rounds, is

1.  $\frac{2{\mathrm{v}}^2}{\mathrm{\pi r}}$

2.  $\frac{4{\mathrm{v}}^2}{3\mathrm{\pi r}}$

3.  $\frac{4{\mathrm{v}}^2}{7\mathrm{\pi r}}$

4.  $\frac{2{\mathrm{v}}^2}{7\mathrm{\pi r}}$

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 ${\mathrm{t}}_2$