The position vector of a particle \(\overrightarrow r\) as a function of time \(t\) (in seconds) is \(\overrightarrow r=3 t \hat{i}+2t^2\hat j~\text{m}\). The initial acceleration of the particle is:
1. \(2~\text{m/s}^2\)
2. \(3~\text{m/s}^2\)
3. \(4~\text{m/s}^2\)
4. zero

A particle is projected on a horizontal surface with kinetic energy K at an angle $\theta$ with the vertical. The kinetic energy of the particle at the highest point of trajectory is:

(1) K

(2) Zero

(3) K ${\cos }^2\mathrm{\theta }$

(4) K ${\sin }^2\mathrm{\theta }$

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:

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?