Relative Motion

8. If the boat crosses in the shortest time, resultant velocity of the boat is \( |\overrightarrow U + \overrightarrow V | = \sqrt {u^2 + V^2 } \)  in magnitude resultant displacement is \( \sqrt {d^2 + x^2 } \)  
Here \( t = \frac{d} {U} = \frac{x} {V} = \frac{{\sqrt {d^2 + x^2 } }} {{\sqrt {U^2 + V^2 } }} \)
9. When the boat crosses the river in the shortest time,the actual path covered is not shortest.The direction of resultant velocity an angle \(\theta\) with the direction of stream such that \( \tan \theta = (u/V) \)

10. To cross the river along the shortest path,boat should be moved along a direction making an angle \( \left( {90^ \circ + \alpha } \right) \) with the stream direction
Here  \( \sin \alpha = \frac{V} {U} \)           
Here time taken by the boat to cross the river is t=\( \frac{d} {{\sqrt {U^2 - V^2 } }} \)   
                
11. When the boat crosses the river along shortest path,time of crossing is not shortest.
Here resultant velocity of boat is \( |\overrightarrow U + \overrightarrow V | = \sqrt {U^2 - V^2 } \) in magnitude.
12.  If t₁ is shortest time to cross the river and  t₂ is the time taken by the boat to travel along the shortest path,then \( \frac{{t_1 }} {{t_2 }} = \frac{{\sqrt {U^2 - V^2 } }} {U} \)