Two particles move from \(A\) to \(C\) and \(A\) to \(D\) on a circle of radius \(R\) and the diameter \(AB.\) If the time taken by both particles is the same, then the ratio of magnitudes of their average velocities is:
                                
1. \(2\)
2. \(2\sqrt{3}\)
3. \(\sqrt{3}\)
4. \(\dfrac{\sqrt{3}}{2}\)

A particle moves on the curve ${\mathrm{}}^{}$\(x^2 = 2y\)$\mathrm{}$. The angle of its velocity vector with the \(x\)-axis at the point \(\left(1, \frac{1}{2}\right )\) will be:

A ball is projected at an angle $\mathrm{\theta }$ with the horizontal. The angle of elevation of the ball when it is at the highest point is:

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

(2)  $\frac{\sin \mathrm{\theta }}{2}$

(3)  $\frac{\cos \mathrm{\theta }}{2}$

(4)  $2 \tan \mathrm{\theta }$

A particle is moving along a curve. Select the correct statement.

A man can row a boat with \(8\) km/h in still water. He is crossing a river of width \(8\) km, where the speed of water flow is \(4\) km/h. What direction should he head the boat to cross the river in the shortest time?
1.  \(30^{\circ}\) with the current
2.  \(60^{\circ}\) with the current
3.  \(90^{\circ}\) with the current
4.  \(120^{\circ}\) with the current

The horizontal range of a particle thrown from the ground is four times the maximum height. The angle of projection with the vertical is:

(1) 60°

(2) 30°

(3) 45°

(4) 90°

A particle is moving 30° north of east with speed 6 m/s. After 3 s the particle is found to be moving along north at the same speed. The magnitude of average acceleration in this interval of time is

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

(2) 6 $\mathrm{m}/{\mathrm{s}}^2$

(3) Zero

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

A particle is thrown with a velocity of 40 m/s. If it passes A and B as shown in the figure at time ${\mathrm{t}}_1$ = 1 s and ${\mathrm{t}}_2$ = 3 s. The value of h is:

(1) 15 m

(2) 10 m

(3) 30 m

(4) 20 m

At a certain moment, the angle between the velocity vector $\overrightarrow{\mathrm{v}}$ and the acceleration $\overrightarrow{\mathrm{a}}$ of a particle is greater than 90°. What can be inferred about its motion at that moment?

(1) It moves along a curve and its speed is decreasing.

(2) It moves along a straight line and accelerated.

(3) It moves along a curve and its speed is increasing.

(4) It moves along a straight line and it is decelerated.

To a stationary man, the rain is falling on his back with a velocity v at an angle $\mathrm{\theta }$ with vertical. To make the rain-velocity perpendicular to the man, he:

(1) must move forward with a velocity vsin$\mathrm{\theta }$.

(2) must move forward with a velocity vtan$\mathrm{\theta }$.

(3) must move forward with a velocity vcos$\mathrm{\theta }$.

(4) should move in the backward direction.