A ship A is moving Westwards with a speed of 10 kmph and a ship B 100 km South of A, is moving Northwards with a speed of 10 kmph. The time after which the distance between them becomes the shortest is:
1. 0 h
2. 5 h
3. $5\sqrt{2}$h
4. $10\sqrt{2}$h

Two particles A and B, move with constant velocities \(\overrightarrow{\mathrm{v}_1}\) and \(\overrightarrow{\mathrm{v}_2}\) respectively. At the initial moment, their position vectors are \(\overrightarrow{\mathrm{r}_1}\) and \(\overrightarrow{\mathrm{r}_2}\) respectively. The condition for particles A and B for their collision will be:

1.\(\frac{\overrightarrow{r_1}-\overrightarrow{r_2}}{\left|\overrightarrow{r_1}-\overrightarrow{r_2}\right|}=\frac{\overrightarrow{v_2}-\overrightarrow{v_1}}{\left|\overrightarrow{v_2}-\overrightarrow{v_1}\right|}\)   
2. \(\overrightarrow{r_1} \cdot \overrightarrow{v_1}=\overrightarrow{r_2} \cdot \overrightarrow{v_2}\)   ${\mathrm{}}_{}$

3. \(\overrightarrow{r_1} \times \overrightarrow{v_1}=\overrightarrow{r_2} \times \overrightarrow{v_2}\)${\mathrm{ \; \; }}_{}$

4. \(\overrightarrow{r_1}-\overrightarrow{r_2}=\overrightarrow{v_1}-\overrightarrow{v_2}\) 

The position vector of a particle R as a function of time t is given by 
$\overrightarrow{\mathrm{R}}=4\sin \left[2\mathrm{\pi t}\right]\hat{\mathrm{i}}+4\cos \left[2\mathrm{\pi t}\right]\hat{\mathrm{j}}$
Where R is in meters, t is in seconds and $\hat{\mathrm{i}},$ $\hat{\mathrm{j}}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of the particle?
 

A projectile is fired from the surface of the earth with a velocity of 5ms^–1 and at an angle θ with the horizontal. Another projectile fired from another planet with a velocity of 3ms^–1 at the same angle follows a trajectory that is identical to the trajectory of the projectile fired from the Earth. The value of the acceleration due to gravity on the other planet is: (given g = 9.8 ms^–2)

1. 3.5 m/s²

2. 5.9 m/s²

3. 16.3 m/s²

4. 110.8 m/s²

The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
              
1. \(-2\hat i+3\hat j~\)
2. \(2\hat i-3\hat j~\)
3. \(2\hat i+3\hat j~\)
4. \(-2\hat i-3\hat j~\)

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is:
1. $\theta ={\tan }^{-1}\left(\frac{1}{4}\right)$
2. $\theta ={\tan }^{-1}\left(4\right)$
3. $\theta ={\tan }^{-1}\left(2\right)$
4. $\theta ={45}^0$

A particle moves in a circle of radius \(5\) cm with constant speed and time period \(0.2\pi\) s. The acceleration of the particle is:
1. \(25\) m/s²
2. \(36\) m/s²
3. \(5\) m/s²
4. \(15\) m/s²

A body is moving with a velocity of \(30\) m/s towards the east. After \(10\) s, its velocity becomes \(40\) m/s towards the north. The average acceleration of the body is:
1. \( 7 \mathrm{~m} / \mathrm{s}^2 \)
2. \( \sqrt{7} \mathrm{~m} / \mathrm{s}^2 \)
3. \( 5 \mathrm{~m} / \mathrm{s}^2 \)
4. \( 1 \mathrm{~m} / \mathrm{s}^2\)

A missile is fired for a maximum range with an initial velocity of 20 m/s. If g= 10 m/s², then the range of the missile will be:

1.  50 m

2.  60 m

3.  20 m

4.  40 m

A projectile is fired at an angle of \(45^\circ\) with the horizontal. The elevation angle \(\alpha\) of the projectile at its highest point, as seen from the point of projection is:
1. \(60^\circ\)
2. \(tan^{-1}\left ( \frac{1}{2} \right )\)
3. \(tan^{-1}\left ( \frac{\sqrt{3}}{2} \right )\)
4. \(45^\circ\)