Relative Motion

MOTION OF BOAT IN A RIVER

1. Let \(\vec U\) be the velocity of boat in still water.is \(\vec V\) velocity of water in a river.If the boat moves in that river,its resultant velocity is \( |\overrightarrow U + \overrightarrow V | = \sqrt {U^2 + V^2 + 2UV\cos \theta } \) (Where \(\theta\) is the angle between U and V)
2. If the boat moves in down stream, \( |\overrightarrow U + \overrightarrow V | = \)U+V( \(\theta\)=0⁰)
    Time taken to cover  a distance d is \( t_1 = \frac{d} {{(U + V)}} \)
3. If the boat moves in up stream, \( |\overrightarrow U + \overrightarrow V | = \)U-V(\(\theta\) =180⁰)
    Time taken to cover  a distance d is \( t_2 = \frac{d} {{(U - V)}} \)
4.  From the above two conditions 
         \( \frac{{t_1 }} {{t_2 }} = \frac{{U - V}} {{U + V}} \Rightarrow \frac{U} {V} = \frac{{t_2 + t_1 }} {{t_2 - t_1 }} \)
5.  If the boat moves at right angles to the stream, \( |\overrightarrow U + \overrightarrow V | = \sqrt {U^2 + V^2 } \)
6. To cross the river from one bank to the other in shortest time, boat should move at right angles  to the stream (along AB)
7. Shortest time to cross the river t=d/U where d is width of the river Here boat suffers drift BC=x
    x=Vt=\( \left( {\frac{V} {U}} \right)d \)