A particle is moving along a circular path of radius a with constant angular velocity $\mathrm{\omega }$. The magnitude of displacement of the particle as a function of time t is:

1.  $\mathrm{a}<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>\mathrm{\omega t}$

2.  $\mathrm{a}<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>\frac{\mathrm{\omega t}}{2}$

3.  $\mathrm{a}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>{\mathrm{\omega t}}^2$

4.  $2\mathrm{a}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\frac{\mathrm{\omega t}}{2}$

In the given diagram shown for a projectile, what is the angle of projection?

1.  ${\tan }^{-1}\left(4\right)$

2.  ${\tan }^{-1}\left(\frac{8}{3}\right)$

3.  ${\tan }^{-1}\left(\frac{4}{3}\right)$

4.  ${\tan }^{-1}\left(\frac{5}{3}\right)$

A body is executing uniform circular motion in a circle of radius 100 m at speed 10 m/s. Acceleration of the body is :

(1) Zero

(2) 2 $\mathrm{m}/{\mathrm{s}}^2$

(3) 1 $\mathrm{m}/{\mathrm{s}}^2$

(4) 5 $\mathrm{m}/{\mathrm{s}}^2$

An object of mass m is projected from the ground with a momentum \(p\) at such an angle that its maximum height is \(\frac{1}{4}\)th of its horizontal range. Its minimum kinetic energy in its path will be:

Which of the following is the angle between velocity and acceleration of a body in uniform circular motion?
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(90^\circ\)

A person is rotating in a horizontal circle of radius 7 m. Find the period of rotation for which the acceleration is equal to 3g.

(1) 2.6 s

(2) 2.8 s

(3) 3.1 s

(4) 4.3 s

A ball is projected horizontally from the top of wall A of height 80 m with speed 10 m/s. At what distance from the wall ball hits the ground?

(1) 20 m

(2) 50 m

(3) 60 m

(4) 40 m

An aeroplane flies \(400\) m north and then \(300\) m west and then flies \(1200\) m upwards. Its net displacement is:

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 $\mathrm{\theta }$

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