A particle starting from the origin \((0,0)\) moves in a straight line in the \((x,y)\) plane. Its coordinates at a later time are ($\sqrt{3}$, \(3)\). The path of the particle makes an angle of __________ with the x-axis:
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(0\)

For a projectile projected at angles (45°-θ) and (45°+θ), the horizontal ranges described by the projectile are in the ratio of:
1. 1:1
2. 2:3
3. 1:2
4. 2:1

A car turns at a constant speed on a circular track of radius \(100\) m, taking \(62.8\) s for every circular lap. The average velocity and average speed for each circular lap, respectively, is:

Two particles A and B are moving in a uniform circular motion in concentric circles of radii \(r_A\) and \(r_B\) with speeds \(v_A\) and \(v_B\) respectively. Their time periods of rotation are the same. The ratio of the angular speed of \(A\)  to that of \(B\) will be:

A particle is projected with a speed u at an angle $\mathrm{\theta }$ to the horizontal. Radius of curvature at highest point of its trajectory is?

1. $\frac{{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{2\mathrm{g}}$

2. $\frac{\sqrt{3}{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{2\mathrm{g}}$

3. $\frac{{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{\mathrm{g}}$

4. $\frac{\sqrt{3}{\mathrm{u}}^2{\cos }^2\mathrm{\theta }}{\mathrm{g}}$

Figure below shows a body of mass M moving with a uniform speed v on a circular path of radius, R. What is the change in acceleration in going from P₁ to P₂? 
              

1. zero

2. ${\mathrm{v}}^2/2\mathrm{R}$

3. $2{\mathrm{v}}^2/\mathrm{R}$

4. $\frac{{\mathrm{v}}^2}{\mathrm{R}}\times \sqrt{2}$

Three balls are thrown from the top of a building with equal speeds at different angles. When the balls strike the ground, their speeds are ${\mathrm{v}}_1,{\mathrm{v}}_2$ $\mathrm{and}$ ${\mathrm{v}}_3$ respectively, then:
           

1. ${\mathrm{v}}_1>{\mathrm{v}}_2>{\mathrm{v}}_3$

2. ${\mathrm{v}}_3>{\mathrm{v}}_2={\mathrm{v}}_1$

3. ${\mathrm{v}}_1={\mathrm{v}}_2={\mathrm{v}}_3$

4. ${\mathrm{v}}_1<{\mathrm{v}}_2<{\mathrm{v}}_3$

A particle moves along a parabolic path y = 9x² in such a way that the x component of the velocity remains constant and has a value of $\frac{1}{3}$ ${\mathrm{ms}}^{-1}$. It can be deduced that the acceleration of the particle will be:

1. $\frac{1}{3}$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

2. $3$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

3. $\frac{2}{3}$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

4. $2$ $\hat{\mathrm{j}}$ ${\mathrm{ms}}^{-2}$

Two particles A and B, move with constant velocities \(\vec{v_1}\)   and  \(\vec{v_2}\) . At the initial moment their position vector are  \(\vec{r_1}\) and  \(\vec{r_2}\)  respectively. The condition for particles A and B for their collision to happen will be:

1.  $\overrightarrow{{\mathrm{r}}_{1 }}.\overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_{2 }}.\overrightarrow{{\mathrm{v}}_2}$

2.  $\overrightarrow{{\mathrm{r}}_1} \mathrm{x} \overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_2} \mathrm{x} \overrightarrow{{\mathrm{v}}_2}$

3. $\overrightarrow{{\mathrm{r}}_1}- \stackrel{\rightharpoonup }{{\mathrm{r}}_2} =\overrightarrow{{\mathrm{v}}_1}- \overrightarrow{{\mathrm{v}}_2}$

4. $\frac{\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}}{\left|\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}\right|}= \frac{\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}}{\left|\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}\right|}$

A missile is fired for a maximum range with an initial velocity of 20 m/s. If g= 10 m/s², then the range of the missile will be:
1. 50 m
2. 60 m
3. 20 m
4. 40 m