Question

You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isnt flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m/s. Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 8.20 m/s . You travel due south from one bank to the other and cross the river in 11.2 s.

With the throttle set so that the speed of the boat relative to the water is 6.00m/s, what is the shortest time in which you could cross the river?

Answer 1

Given details :

Speed of boat while swimming north = v₁ = 6.00 m/s

Speed of boat while swimming south = v₂ = 8.20 m/s

Time taken swimming north = t₁ = 20.1 s

Time taken swimming south = t₂ = 11.2 s

Assumptions :

Let the speed of river flowing east = v_r m/s

Let the width of river = d m

Thus, resultant velocity while swimming north V_N (say) = √ (v₁² + v_r²) = √ (36 + v_r²) m/s

Resultant velocity while swimming north V_S (say) = √ (v₂² + v_r²) = √ (67.24 + v_r²) m/s

Width of river = d = V_N * t₁ = V_S * t₂ [ as, distance = speed * time]

Or, √ (36 + v_r²) * 20.1 = √ (67.24 + v_r²) * 11.2

Or, (36 + v_r²) * 404.01 = (67.24 + v_r²) * 125.44

Taking modulus value, we have v_r²  = (6109.7744 / 278.57) m²/s² = 21.93 m²/s²

Or, v_r = 4.683 m/s

Thus, width of river d = √ (36 + v_r²) m/s * 20.1 s

= √ (36 + 21.93) m/s * 20.1 s = 152.98 m

Now to calculate shortest time for crossing the river, the horizontal component of velocity should be zero, so that there is no deflection in the boat while crossing and the distance covered is exactly the perpendicular distance across the river.

Thus, fixing speed of boat relative to water = v_B (say) = 6.00 m/s, we must have velocity of river v_r = 0 so as to nullify the horizontal component.

Thus, minimum time = 152.98 m / (6.00 m/s) = 25.497 s = 25.5 s (approx)