A particle is thrown at an angle of projection $\mathrm{\theta }$ = 45° with speed u. The average velocity of the particle during its ground to ground flight is

1.  $\sqrt{2}\mathrm{u}$

2.  $\frac{\mathrm{u}}{\sqrt{2}}$

3.  $\frac{\mathrm{u}}{2}$

4.  0

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 bomb is dropped from an aeroplane flying horizontally. The path of the bomb as seen by the pilot will be (neglect air friction)

(1) A straight line

(2) A parabola

(3) An ellipse

(4) A hyperbola

The range of a bullet fired from a gun at an angle $\left(\frac{\mathrm{\pi }}{4} - \mathrm{\varphi }\right)$ with the horizontal is the same as the range of another bullet fired from that gun at an angle $\theta$ with the horizontal. Then,

1.  $\mathrm{\theta } = \mathrm{\varphi }$

2.  $\mathrm{\theta } = \frac{\mathrm{\pi }}{4}$

3.  $\mathrm{\theta } = \frac{\mathrm{\pi }}{4} + \mathrm{\varphi }$

4.  $\mathrm{\theta } = \mathrm{\varphi } - \frac{\mathrm{\pi }}{4}$

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.  $\sqrt{2}\mathrm{R}$

Two projectiles P and Q are projected with the same speed at angles 60° and 30° with horizontal on level ground, then: [R = Range, T = Time of flight, H = Maximum height]

(1) ${\mathrm{R}}_{\mathrm{P}}=2{\mathrm{R}}_{\mathrm{Q}}$

(2) ${\mathrm{T}}_{\mathrm{P}}=\sqrt{3} {\mathrm{T}}_{\mathrm{Q}}$

(3) $\mathrm{H}=3{\mathrm{H}}_{\mathrm{P}}$

(4) All of these

A body is projected with the same speed at two different angles covers the same horizontal distance R. If ${\mathrm{T}}_1 \mathrm{and} {\mathrm{T}}_2$ are two times of flights, then R is equal to :

1.  $\frac{{\mathrm{gT}}_1{\mathrm{T}}_2}{2}$

2.  $2{\mathrm{gT}}_1{\mathrm{T}}_2$

3.  $\frac{{\displaystyle \frac{1}{2}{\mathrm{gT}}_1{\mathrm{T}}_2}}{\left({\mathrm{T}}_1 + {\mathrm{T}}_2\right)}$

4.  $\sqrt{2}{\mathrm{gT}}_1{\mathrm{T}}_2$

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

A particle is revolving in a circular path of radius 200 m at a speed of 20 m/s. Its speed is increasing at a rate of $\sqrt{5}$ $\mathrm{m}/{\mathrm{s}}^2$. Its acceleration is :

(1) 2 $\mathrm{m}/{\mathrm{s}}^2$

(2) $\sqrt{7}$ $\mathrm{m}/{\mathrm{s}}^2$

(3) 3 $\mathrm{m}/{\mathrm{s}}^2$

(4) $\sqrt{5}$ $\mathrm{m}/{\mathrm{s}}^2$