A projectile is projected from the ground with the velocity ${\mathrm{v}}_0$ at an angle $\theta$ with the horizontal. What is the vertical component of the velocity of the projectile when its vertical displacement is equal to half of the maximum height attained?

1.  $\sqrt{3}{\mathrm{v}}_0$ $\mathrm{cos\theta }$

2.  $\frac{{\mathrm{v}}_0}{\sqrt{2}}\sin$ $\mathrm{\theta }$

3.  $\frac{{\mathrm{v}}_0}{\sqrt{2}}\cos \theta$

4.  $\sqrt{5}{\mathrm{v}}_0$

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 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. $\frac{\mathrm{\theta }}{\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

2. $\frac{\mathrm{\theta }}{2\sin {\displaystyle \frac{\mathrm{\theta }}{2}}}$

3. $\frac{\mathrm{\theta }}{2\cos {\displaystyle \frac{\mathrm{\theta }}{2}}}$

4. $\frac{\mathrm{\theta }}{{\displaystyle \cos \frac{\mathrm{\theta }}{2}}}$

Two particles move from A to C and A to D on a circle of radius R and diameter AB. If the time taken by both particles are the same, then the ratio of magnitudes of their average velocities is:
                                
1. 2

2.  $2\sqrt{3}$

3.  $\sqrt{3}$

4.  $\frac{\sqrt{3}}{2}$

A particle moves on the curve ${\mathrm{}}^{}$\(x^2 = 2y\)$\mathrm{}$. The angle of its velocity vector with the x-axis at the point \(\left(1, \frac{1}{2}\right )\) will be:

A particle is moving along a curve. Select the correct statement.

A particle starts moving from the origin in the XY plane and its velocity after time \(t\) is given by \(\overrightarrow{\mathrm{v}}=4 \hat{\mathrm{i}}+2 \mathrm{t} \hat{\mathrm{j}}\). The trajectory of the particle is correctly shown in the figure:

1.		2.
3.		4.

A particle is moving in the XY plane such that \(x = \left(t^2 -2t\right)\) m, and \(y = \left(2t^2-t\right)\) m, then:

Path of a projectile with respect to another projectile so long as both remain in the air is:

1. Circular

2. Parabolic

3. Straight

4. Hyperbolic

A particle is moving along a circle of radius \(R \) with constant speed \(\mathrm{v}_0\). What is the magnitude of change in velocity when the particle goes from point \(A\) to \(B \) as shown?