The figure shows a rod of length \(5\) m. Its ends, \(A\) and \(B\), are restrained to moving in horizontal and vertical guides. When the end \(A\) is \(3\) m above \(O\), it moves at \(4\) m/s. The velocity of end \(B\) at that instant is:
                         
1. \(2\) m/s
2. \(3\) m/s
3. \(4\) m/s
4. \(0.20\) m/s

What is the velocity of the block when the angle between the string and the horizontal is \(30^\circ\) as shown in the diagram?

  
1. \(v_B=v_P\)
2. \(v_B=\frac{v_P}{\sqrt{3}}\)
3. \(v_B=2v_P\)
4. \(v_B=\frac{2v_P}{\sqrt{3}}\)

If the block is being pulled by the rope moving with speed \(v\) as shown, then the horizontal velocity of the block is: