A projectile is projected with initial kinetic energy \(K\). If it has kinetic energy \(0.25K\) at its highest point, then the angle of projection is:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(75^{\circ}\)

A body is projected from the ground at an angle of 30° with horizontal. The instantaneous velocity of the body becomes perpendicular to initial velocity during its motion:

(1) Once

(2) Twice

(3) Thrice

(4) Never

In the uniform circular motion, which of the following physical quantity remains constant?

(1) Angular velocity

(2) Angular momentum

(3) Angular speed

(4) All of these

A body is projected with velocity $\overrightarrow{\mathrm{v}}=<mspace\; width="1em"></mspace>\left(\mathrm{\alpha }\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\mathrm{\beta }\hat{\mathrm{j}}\right)$ m/s. The time of flight of body considering x as horizontal and y as the vertical axis and g is the acceleration due to gravity:

1.  $\frac{2\mathrm{\beta }}{\mathrm{g}}$

2.  $\frac{2\mathrm{\alpha }}{\mathrm{g}}$

3.  $\frac{2\mathrm{\alpha \beta }}{\mathrm{g}}$

4.  $\frac{2\mathrm{\alpha }}{\mathrm{g\beta }}$

Two balls are projected along the same initial line from a height h = 20 m. The direction of the projections is horizontal. If initial speeds are 20 m/s and 25 m/s, the maximum separation between the balls when they will hit the ground is

(1) 40 m

(2) 50 m

(3) 10 m

(4) 90 m

If the position of a particle varies according to the equations \(x= 3t^2\)${\mathrm{}}^{}$, \(y =2t\), and \(z= 4t+4\), then which of the following is incorrect?

If three coordinates of a particle change according to the equations \(x = 3 t^{2}, y = 2 t , z= 4\), then the magnitude of the velocity of the particle at time \(t=1\) second will be:
1. \(2\sqrt{11}~\text{unit}\)
2. \(\sqrt{34}~\text{unit}\)
3. \(40~\text{unit}\)
4. \(2\sqrt{10}~\text{unit}\)

Two bodies of equal masses are projected, one from top of the tower in the horizontal direction and other from the foot of tower at an angle of 45° with the horizontal, then the acceleration of their center of mass is

(1) $\frac{\mathrm{g}}{\sqrt{2}}$downward

(2) g downward

(3) $\frac{\mathrm{g}\left(\sqrt{2}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>1\right)}{\sqrt{2}}$ downward

(4) $\frac{\mathrm{g}\left(\sqrt{2}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>1\right)}{\sqrt{2}}$ upward

A projectile is projected at an angle of 60° from horizontal. At an instant, it is moving at an angle of 30° with horizontal with velocity 10 m/s. The radial acceleration of the particle at that instant is:

1.  $\frac{20}{\sqrt{2}}\mathrm{m}/{\mathrm{s}}^2$

2.  $\frac{20}{\sqrt{3}}\mathrm{m}/{\mathrm{s}}^2$

3.  $20\sqrt{3}\mathrm{m}/{\mathrm{s}}^2$

4.  $5\sqrt{3}\mathrm{m}/{\mathrm{s}}^2$

The initial velocity of a projectile is given by $\mathrm{v}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left(20\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>50\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>70\hat{\mathrm{k}}\right)$ m/s. If acceleration due to gravity is g = 10($-\hat{\mathrm{k}}$) m/${\mathrm{s}}^2$, then the time after which projectile returns to the same horizontal level is:

(1) 4 s

(2) 10 s

(3) 14 s

(4) 12 s