A particle is rotating with increasing speed on a circular track. The angle between its radius vector and acceleration is $\theta$, then:

(1) 0° < $\mathrm{\theta }$ < 90°

(2) 45° < $\mathrm{\theta }$ < 90°

(3) 90° < $\mathrm{\theta }$ < 180°

(4) $\mathrm{\theta }$ = 90°

A particle starts moving on a circular path from rest, such that its tangential acceleration varies with time as $a_t=kt$. Distance traveled by particle on the circular path in time $t$ is:
1. $\frac{kt^3}{3}$
2. $\frac{kt^2}{6}$
3. $\frac{kt^3}{6}$
4. $\frac{k t^2}{2}$

A projectile thrown from the ground has horizontal range R. If velocity at the highest point is doubled somehow, the new range will be:

(1) 3 R

(2) 1.5 R

(3) $\sqrt{3}$ R

(4) 2 R

The time of flights for a particle thrown for equal range at different angles is 4 sec and 3 sec. Speed of projection is:

(1) 10 m/s

(2) 20 m/s

(3) 25 m/s

(4) 50 m/s

A body is projected from the ground at a speed v at an angle $\mathrm{\theta }$ above the horizontal. The radius of curvature of its path at the highest point is:

(1)  $\frac{{\mathrm{v}}^2 {\cos }^2 \mathrm{\theta }}{2\mathrm{g}}$

(2)  $\frac{\mathrm{v} \cos \mathrm{\theta }}{\mathrm{g}}$

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

(4)  $\frac{{\mathrm{v}}^2 {\cos }^2 \mathrm{\theta }}{\mathrm{g}}$

A particle is projected from the origin with velocity $\left(4\hat{\mathrm{i}} + 9\hat{\mathrm{j}}\right)$ 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 is moving with a constant speed of 20 m/s on a horizontal road. Its motion is recorded by a camera, which is at 40 m from the road as shown. In order to focus the car, the camera is to be rotated at an angular speed equal to :

(1) 1.000 rad/s

(2) 0.500 rad/s

(3) 0.250 rad/s

(4) 0.125 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. $\dfrac{\theta}{\cos\frac{\theta}{2}}$