A particle is projected from the origin with velocity $\left(4\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>9\hat{\mathrm{j}}\right)$ m/s. The acceleration in the region is constant and -10$\hat{\mathrm{j}}$ $\mathrm{m}/{\mathrm{s}}^2$. The magnitude of velocity after one second is

(1) 8 m/s

(2) $\sqrt{15}$ m/s

(3) $\sqrt{17}$ m/s

(4) $\sqrt{27}$ m/s

The speed of a boat in still water is half of the velocity of the flow of water(v). At what angle boat should steer with the direction of flow of water so that drift of boat is minimum?

(1) 30°

(2) 60°

(3) 90°

(4) 120°

A sports car travels at \(20~\text{m/s}\) along a horizontal road. A camera positioned \(40~\text{m}\) horizontally away from the road (as shown in the diagram) tracks the car at a \(45^\circ\) viewing angle. What angular velocity must the camera rotate at to keep the car in focus?

1. \(1.000~\text{rad/s}\)
2. \(0.500~\text{rad/s}\)
3. \(0.250~\text{rad/s}\)
4. \(0.125~\text{rad/s}\)

The magnitude of a vector $\overrightarrow{\mathrm{A}}$ is constant but it is changing its direction continuously. The angle between $\overrightarrow{\mathrm{A}}$ and $\frac{\mathrm{d}\overrightarrow{\mathrm{A}}}{\mathrm{dt}}$ is :

(1) 180°

(2) 120°

(3) 90°

(4) 0°

A particle is moving on a circular path of radius \(R.\) When the particle moves from point \(A\) to \(B\) (angle \( \theta\)), the ratio of the distance to that of the magnitude of the displacement will be:

         
1. \(\dfrac{\theta}{\sin\frac{\theta}{2}}\)
2. \(\dfrac{\theta}{2\sin\frac{\theta}{2}}\)
3. \(\dfrac{\theta}{2\cos\frac{\theta}{2}}\)
4. \(\dfrac{\theta}{\cos\frac{\theta}{2}}\)

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

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

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

(4)  $2<mspace\; width="1em"></mspace>\tan <mspace\; width="1em"></mspace>\mathrm{\theta }$

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