Raindrops are falling with a speed v vertically downwards and a man is running on a horizontal road with speed u. The magnitude of the velocity of the raindrops with respect to the man is:
1. v - u
2. v + u
3. \(\sqrt{\text{v}^2 + \text{u}^2 \over 2}\)
4. \(\sqrt{\text{v}^2 + \text{u}^2}\)

A boy runs on a circular track of radius \(\mathrm{R}\) (in km) with a speed of $\frac{\mathrm{\pi R}}{2}$ km/h in the clockwise direction for \(3\) h and then with $\mathrm{\pi R}$ km/h in the anticlockwise direction for \(1\) h. The magnitude of his displacement will be: 
1.  $\frac{\mathrm{\pi R}}{2}$

2.  $\frac{\mathrm{R}}{\sqrt{2}}$

3.  $\frac{3\mathrm{\pi R}}{2}$

4.  $\sqrt{2}\mathrm{R}$

The equation of trajectory of a projectile is given by y = x - 10${\mathrm{x}}^2$. Its speed of projection is: (g = 10 m/${\mathrm{s}}^2$)

1. 1 m/s

2. 2 m/s

3. 3 m/s

4. 4 m/s

A body starts moving from rest on a horizontal ground such that the position vector of the body with respect to its starting point is given by $\mathrm{r}$ $=$ $2\mathrm{t}\hat{\mathrm{i}}$ $+$ $3{\mathrm{t}}^2\hat{\mathrm{j}}$. The equation of the trajectory of the body is:

1. y = 1.5 x

2. y = 0.75 ${\mathrm{x}}^2$

3. y = 1.5 ${\mathrm{x}}^2$

4. y = 0.45 ${\mathrm{x}}^2$

A particle starts moving with constant acceleration with initial velocity (\(\hat{\mathrm{i}}+5\hat{\mathrm{j}}\)$\stackrel{}{\mathrm{}}$) m/s. After \(4\) seconds, its velocity becomes (\(3\hat{\mathrm{i}}-2\hat{\mathrm{j}}\)) m/s. The magnitude of its displacement in 4 seconds is:
1. \(5\) m
2. \(10\) m
3. \(15\) m
4. \(20\) m

If a particle is moving in a circular orbit with constant speed, then:

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°

2. 45°

3. 60°

4. 75°

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

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

In a uniform circular motion, if the speed of the particle is 2 m/s and radius of the circle is 2 m, then the values of centripetal and tangential acceleration are, respectively:

1. 2 $\mathrm{m}/{\mathrm{s}}^2$, 2 $\mathrm{m}/{\mathrm{s}}^2$

2. 2 $\mathrm{m}/{\mathrm{s}}^2$, 1 $\mathrm{m}/{\mathrm{s}}^2$

3. 0, 2 $\mathrm{m}/{\mathrm{s}}^2$

4. 2 $\mathrm{m}/{\mathrm{s}}^2$, 0