Question

on a calm a boat can go across a lake and return in time T0 (knot) at a speed V. on a rough day there is a uniform current at speed v to help the onward journey and impede the return journey. if the time taken to go across and return on the rough day be T, then T/T0 will be ?

Answer 1

on a calm a boat can go across a lake and return in time T0 (knot) at a speed V. on a rough day there is a uniform current at speed v to help the onward journey and impede the return journey. if the time taken to go across and return on the rough day be T, then T/T0 will be ?

Solution:

Let the distance of the trip (go across and return) is D, then the travel time in calm day is:

$$T _ {0} = \frac {D}{V} + \frac {D}{V} = \frac {2 D}{V}$$

the travel time in rough day is:

$$T = \frac {D}{V + v} + \frac {D}{V - v} = \frac {D (V - v + V + v)}{(V + v) (V - v)} = \frac {2 D V}{(V + v) (V - v)}$$ $$(2) \div (1):$$ $$\frac {T}{T _ {0}} = \frac {2 D V}{(V + v) (V - v)} \cdot \frac {V}{2 D} = \frac {V ^ {2}}{V ^ {2} - v ^ {2}} = 1 - \frac {V ^ {2}}{v ^ {2}}$$

Answer: $\frac{T}{T_0} = 1 - \frac{V^2}{v^2}$