Question

An athlete needs to get across a river. She must reach from point A on one bank to point B, which is directly across from point A on the opposite bank. If she wishes to minimize the total time T it would take to do so, at what angle upstreams (measured from the line AB) should she swim? Let her swimming/rowing speed relative to the water be 2.0 mi/h, and her running speed along the bank be 5 mi/h. The river flows downstream at 1.5 mi/h. (The rule of the competition stipulates that the athletes must swim first from point A, then run to reach point B.)

Answer 1

Answer:

θ = 90°

Explanation:

Speed at which she can swim in stationary water; v_m = 2 mi/h

Speed at which the river is flowing downstream; v_r = 1.5 mi/h

This can be illustrated in a form of triangle as attached. Where θ is the angle to the x-axis.

With respect to the bank of the river, the total speed of the girl is;

V_R = v_r + v_m

Now the horizontal component is;

V_R = v_r + v_m(cos θ)

While the vertical component is;

V_R = v_m(sin θ)

Now, the total time taken to get across the river is;

T = width of river/vertical component of total speed

Let's denote width of river as d.

Thus;

T = d/(v_m(sin θ))

Now, we are told she wishes o minimize the total time(T) .

Since the width(d) of the river is constant, and v_m is fixed, then it means the only variable we have is the angle θ.

Now, for the total time to be minimized, the denominator has to be at maximum.

Since angle θ is the only one that is variable, we have to find the value of angle θ that will make sin θ to be at a maximum value.

Now, for sin θ to be maximum, it means it has to be equal to 1.

Thus, θ = sin^(-1) 1

θ = 90°