Question

Two boat landings are 2.7 km apart on the same bank of a stream that flows at 2.5 km/hr. A motorboat makes the round trip between the two landings in 2.9 hr. What is the speed of the boat relative to the water? Answer in units of km/hr.

please explain in detail step by step.

Answer 1

To resolve this type of exercises use the theorem of relative speed, in one dimension is relatively easy

V_BE = V_Bw + V_WE        bold symbol vector

The subscripts indicate

B   boat

W   water

E     Earth

You can see that the internal indices (W) are cancelled out in the sum.

In the first part of the journey the boat and the water goes in the same direction and to the second part of the journey is going in the opposite direction.

V_Bw + V_WE = d/t1                       1

V_Bw - V_WE = d/t2                        2

. t = t1 + t2

It should be noted that the times are not equal

We solve the time den equations 1 and 2, then we replace in the total time

. t1 = d /( V_Bw + V_WE )

. t2= d/( V_Bw - V_WE )

. t = d /( V_Bw + V_WE ) + d/( V_Bw - V_WE )

We do the algebra for calculated V_BW

. t/d =1 /( V_Bw + V_WE ) + 1/( V_Bw - V_WE )

. t/d = ( ( V_Bw - V_WE ) + ( V_Bw + V_WE ) )/ ( (V_Bw + V_WE) (V_Bw - V_WE ) )

Observation :   ( (V_Bw + V_WE) (V_Bw - V_WE ) ) = V_Bw² + V_BW V_WE -V_BWV_WE   - V_WE²)

. t/d = 2 V_Bw /( V_Bw² - V_WE²)

. V_Bw² - V_WE² = (2d/t) V_BW

We write and solve this quadratic equation

V_Bw² - (2d/t) V_BW   - V_WE² =0                       2d/t= 2 2.7/2.9 =1.86

V_Bw² - 1.86 V_BW   - 6.25 = 0

V_BW = 1.86 ±Ö (1.86² – 4 (-6.25)) / 2 =1.86 ±Ö (28.4596)) / 2

V_BW = (1.86 ± 5.33)/2

results

1. V_BW =3.595 Km/h
2. V_BW = -1.735 Km/h

We must analyze these two results to see if some gives a non-acceptable value. There are several ways to do this.

.1      The first is to replace the value in the value gives and see which meets the condition that total time 2.9, in this case we see that 3.595 Km/h speed satisfies

.2       The second way is to see that negative speed makes no sense since the pot reverse in the river and can not reach destination

In my personal opinion, the first method is the most secure