Four particles lie initially at the corners of a square of side length L. All the particles start to move with speed v. A moves towards B, B moves towards C, C moves towards D and D moves towards A. The distance covered by a particle till they meet is

                                 

(1) $\frac{\mathrm{L}}{\sqrt{5}}$

(2) L

(3) $\sqrt{2}\mathrm{L}$

(4) 2L

A boat can move with a maximum speed of 10 m/s in still water. If the speed of river water is 5 m/s, then in how much minimum time the boat can cross the river of width 500 m?

(1) $\frac{100}{\sqrt{5}}$ s

(2) 50 s

(3) 100 s

(4) 150 s

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 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. \(\sqrt{{v}^2 + {u}^2 \over 2}\)
4. \(\sqrt{{v}^2 + {u}^2}\)

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 \(R\) (in km) with a speed of \(\frac{πR}{2}\) km/h in the clockwise direction for \(3\) h and then with \(πR\) km/h in the anticlockwise direction for \(1\) h. The magnitude of his displacement will be: 
1. \(\frac{πR}{2}\)
2. \(\frac{R}{\sqrt{2}}\)
3. \(\frac{3πR}{2}\)
4. \(\sqrt{2}R\)

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 \(r= 2 t\hat{i}+3t^2\hat j\). The equation of the trajectory of the body is:
1. \(y =1.5x\)
2. \(y =0.75x^2\)
3. \(y =1.5x^2\)
4. \(y =0.45x^2\)