Motion in 2 Dimensions & Relative velocity

[EEristavi]

Homework Statement

The water in a river flows uniformly at a constant speed
of 2.50 m/s between parallel banks 80.0 m apart. You
are to deliver a package across the river, but you can
swim only at 1.50 m/s.
(c) If you choose to minimize the distance downstream
that the river carries you, in what direction should you
head? (d) How far downstream will you be carried?

Homework Equations

S = V T
V = V₁ + V₂

The Attempt at a Solution

V = 1.5 - swimmer's velocity
L = 80 - distance between banks
V_r = 2.5 -Velocity of river
α - Angle between distance between banks and velocity
t - time required to cross the river
d - distance traveled downV cosα t = L (1)
(V_r - V sinα) t = d (2)(1) ->
t = L / V cosα (3)(2), (3) ->
(V_r - V sinα) L / (V cosα) = d

V_r L secα / V - V L tanα = d

(Find derivative and set equal to 0 - to find extreme point)
V_r L secα tanα/ V - V L sec²α = 0

133 secα tanα - 120 sec²α = 0

secα (133 tanα - 120 secα) = 0

133 tanα =120 secα

sinα = 120/133

α ≅ 64Note: In solutions, I read that the α ≅ 53.1.
Moreover, it says that resultant velocity must be perpendicular to swimming velocity (but it's not written why)

Can you tell what I'm doing wrong and why it must be perpendicular (it seems like this is the starting point for the solution)

 

[DrClaude]

(2), (3) ->
(V_r - V sinα) L / (V cosα) = d

V_r L secα / V - V L tanα = d

Second line above is incorrect.

 

[EEristavi]

Second line above is incorrect.

I see... I had other problem and when I fixed it, I made this one :(

Sorry for such a lame question - Though it was something conceptual.
Thank you.

Can you tell me something about perpendicularity?

 

[Orodruin]

Here is a diagram (click for larger size):

The green arrow is the river flow velocity $\vec v_{\rm river}$, the red arrows represent possible swimming velocities, which must all be within the red circle. The blue arrows therefore represent different possible resultant velocities. The direction of the resultant velocity will tell you how far downstream you will be carried. Clearly, the vector with the least steep tilt (which will mean the shortest downstream distance) occurs when the resultant velocity is tangent to the red circle, which will mean that it is orthogonal to the circle's radius, i.e., the corresponding swimming velocity.

Edit: This is the geometrical reason. You should of course dress this in mathematical terms yourself.

 

[EEristavi]

@Orodruin Nicely explained! Thanks