# Riverâ€“boat or riverâ€“man problem

Let a boat can travel in still water with a velocity

*v*. If water also moves in the river, then net velocity of boat (wrt ground) will be different from*v*.Net velocity of the boat: where is velocity of boat in water or wrt water and is the velocity of water in river.

# Special cases

- If boat travels downstream:
**Fig. 3***v*+*u*) downstream. - If boat travels upstream
**Fig. 4***v*â€“*u*upstream. - If boat travels at some angle
*Î¸*with downstream:**Fig. 5***t*=*x*= (*u*+*v*cos*Î¸*)*t*â‡’*x*=

# To cross the river in shortest time

sin

*Î¸*= 1 â‡’*Î¸*= 90Â°,*t*_{min}Â = ,*x*=**Fig. 6**

*x*can be calculated from Fig. 6 also:

**tan**

*Î±*= â‡’

*x*=

Net velocity of boat (put

*Î¸*= 90Â°), ,Magnitude,

# To cross the river by shortest path

For this

*x*= 0 â‡’*u*+*v*cos*Î¸*= 0**Fig. 7**

â‡’ cos

*Î¸*= â€“ â‡’ cos(90 +*Î²*) = â€“ â‡’ sin*Î²*=Net velocity of boat: ,

Magnitude,

*=*_{ }=# Rainâ€“man problem

Formula to be applied: where is velocity of rain wrt man, is the velocity of rain (wrt ground) and is the velocity of man (w.r.t. ground).

**Case I:**Rain is falling vertically downwards with velocity

*v*and

_{r}*a*

*man is running horizontally with velocity*

*v*as shown in Fig. 8. What is the relative velocity of rain wrt man?

_{m}**Fig. 8**

Given: ,

Now

or

**Fig. 9**

**Magnitude:**,

**Direction:**tan

*Î±*=

**Case II:**If rain is already falling at some angle

*Î¸*with horizontal (Fig. 10), then with what velocity the man should travel so as to him the rain appears falling vertically downwards?

**Fig. 10**

Here:

Now = (

*v*sin_{r}*Î¸*â€“*v*)_{m}Now for rain to appear falling vertically, the horizontal component of should be zero, i.e.,

*v*sin

_{r}*Î¸*â€“

*v*= 0 â‡’ sin

_{m}*Î¸*= and

*v*

_{r}_{/m}=

*v*

_{r}_{ }cos

*Î¸*

*= v*

_{r}**Fig. 11**