## Motion in a Plane

# Relative Velocity inTwo dimension

- Rain is falling with velocity \overline{V_{R}} vertically down , man is moving with velocity \overline{V_{Y}} then velocity of rain with respect to man V = \overline{V_{r}} - \overline{V_{y}}
- The angle with the vertical to hold umbrella to protect himself from rain tan α = V
_{m}/V_{r}where V_{m}is velocity of man 'V_{r}' is the velocity of rain. - Velocity of 'A' with respect to "B" is V_{AB} = \sqrt{VA^{2} + VB^{2} - V_{A}V_{B} \cos \theta} where V
_{A}is velocity of A 'V_{B}' is Velocity of 'B' - Time taken by boat to cross river in shortest path is t = \frac{d}{v_{b}} where 'd' is the width of the river V
_{b}is velocity of boat - To cross the river in shortest path boat should be sowed at angle 90 + θ with the river flow where θ = sin
^{−1}(V_{r}/V_{br}) (v_{r}= Velocity of river v_{br}= Resultant velocity of boat)

### View the Topic in this video From 0:03 To 6:11

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1. If the swimming is in the direction of flow of water, *v _{m}* =

*v*+

*v*

_{R}2. If the swimming is opposite to the flow of water, *v _{m}* =

*v*−

*v*

_{R}3. If rain is falling vertically with a velocity \vec{v}_{R} and an observer is moving horizontally with speed \vec{v}_{M} the velocity of rain relative to observer will be \vec{v}_{RM} = \vec{v}_{R} - \vec{v}_{M}

4. Which by law of vector addition has magnitude v_{RM} = \sqrt{v_{R}^{2} + v_{M}^{2}}

5. Direction \theta = \tan^{-1} (v_{M}/v_{R})

6. Time taken for down stream = \tt \frac{width \ of \ river}{v_{m} + v_{R}}

7. Time taken for up stream = \tt \frac{width \ of \ river}{v_{m} - v_{R}}