Velocity and acceleration vectors of a particle moving on a circular path are shown here. At the instant shown:

                            

(1) The speed of the particle must be decreasing.

(2) Acceleration of particle must be increasing in magnitude.

(3) The dot product of $\overrightarrow{\mathrm{v}}$ and $\overrightarrow{\mathrm{a}}$ is positive.

(4) The magnitude of the momentum of the particle must be increasing in magnitude.

A cricketer can throw a ball to a maximum horizontal distance of 50 m. How much high above the ground can he throw the same ball?

(1) 50 m

(2) 25 m

(3) 75 m

(4) 100 m

Coordinates of a particle as a function of time t are x = 2t, y = 4t. It can be inferred that the path of the particle will be:

1. Straight line

2. Ellipse

3. Parabola

4. Hyperbola

When a man walks on a horizontal road with velocity 1 km/h, the rain appears to him coming vertically at a speed of 2 km/h. The actual speed of the rain with respect to ground is:

1. $\sqrt{3}$ km/h

2. $\sqrt{5}$ km/h

3. 1 km/h

4. 3 km/h

Path of a projectile as seen from another projectile :

(1) Straight

(2) Circular

(3) Hyperbolic

(4) Parabolic

At an instant, the velocity and acceleration of a particle moving in the XY plane are $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ m/s and $-3\hat{\mathrm{i}}+2\hat{\mathrm{j}}$ $\mathrm{m}/{\mathrm{s}}^2$. Rate of change of speed of the particle at this instant is:

(1) $\sqrt{13}$ $\mathrm{m}/{\mathrm{s}}^2$

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

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

(4) Zero

Which of the following statement/s is/are incorrect regarding the motion in a plane?

A body started moving with an initial velocity of \(4\) m/s along the east and an acceleration \(1\) m/s² along the north. The velocity of the body just after \(4\) s will be?

The radius vector of a particle moving on a circle is given by $\overrightarrow{\mathrm{r}} = \mathrm{AcosBt}\hat{\mathrm{i}} + \mathrm{AsinBtj}\stackrel{}{^}$ (A and B are constants). The radius of the circle and speed of the particle, respectively, are

1.  A, AB

2.  $\mathrm{A}, \frac{{\mathrm{A}}^2}{\mathrm{B}}$

3.  B, AB

4.  $\mathrm{B}, \frac{{\mathrm{A}}^2}{\mathrm{B}}$

For a particle projected on a horizontal surface, the ratio of range to maximum height is:

($\mathrm{\theta }$ is the angle of projection with respect to horizontal):

(1) 4cot$\mathrm{\theta }$

(2) 4tan$\mathrm{\theta }$

(3) 0.25cot$\mathrm{\theta }$

(4) 0.25sin$\mathrm{\theta }$