The position of a particle is given by; \(\vec{r}=(3.0t\hat{i}-2.0t^{2}\hat{j}+4.0\hat{k})\) m $\stackrel{}{\mathrm{}}$
 where \(t\) is in seconds and the coefficients have the proper units for \(r\) to be in meters. The magnitude and direction of \(\vec{v}(t)\) at \(t=1.0\) s are:

1.	\(4\) m/s \(53^\circ\) with x-axis
2.	\(4\)  m/s \(37^\circ\) with x-axis
3.	\(5\) m/s \(53^\circ\) with y-axis
4.	\(5\)  m/s \(53^\circ\) with x-axis

An insect trapped in a circular groove of radius \(12\) cm moves along the groove steadily and completes \(7\) revolutions in \(100\) s. What is the angular speed of the motion? 
1.\(0.62\) rad/s
2.\(0.06\) rad/s
3. \(4.40\) rad/s
4. \(0.44\) rad/s

Two particles having mass \(M\) and \(m\) are moving in a circular path having radius \(R\) & \(r\) respectively. If their time periods are the same, then the ratio of angular velocities will be: 
1. \(\frac{r}{R}\)
 2. \(\frac{R}{r}\)
3. \(1\)
4. \(\sqrt{\frac{R}{r}}\)

If two projectiles, with the same masses and with the same velocities, are thrown at an angle \(60^\circ\) & \(30^\circ\) with the horizontal, then which of the following quantities will remain the same?

Let \(\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}},\) then:

In a projectile motion the velocity,

Let \(\mathrm{ABCDEF}\) be a regular hexagon, with the vertices taken in order. The resultant of the vectors: \(\overrightarrow{AB},~\overrightarrow{BC},~\overrightarrow{CD},~\overrightarrow{DE}\) equals, in magnitude, the vector:
1. \(\overrightarrow{AB}\)
 2. \(\overrightarrow{AD}\)
3. \(\sqrt2\overrightarrow{AB}\)
 4. \(\sqrt3\overrightarrow{AB}\)

Two projectiles are launched, one at twice the speed of the other; the slower one at \(30^\circ\) and the faster one at \(60^\circ.\) Their horizontal ranges are in the ratio: (slower : faster)
1. \(\frac{1}{2}\)
2. \(\frac{1}{4}\)
3. \(\frac{1}{6}\)
4. \(\frac{1}{12}\)

For a particle performing uniform circular motion,

Choose the correct statement/s:

In a two-dimensional motion, instantaneous speed ${\mathrm{v}}_0$ is a positive constant. Then which of the following is necessarily true?