What angle should a swimmer aim to land directly across the starting position?

[WorkingHard2017]

Homework Statement

A swimmer who achieves a speed of of 0.75 m/s in still water swims directly across a river 72m wide. The swimmer lands on the far shore at a position 54m downstream from the starting point. Determine the direction the swimmer would have the aim to land directly across from the starting position

Homework Equations

v swimmer relative to ground = v swimmer relative to water + v water relative to ground

The Attempt at a Solution

v water relative to ground = 0.56 m/s[/B]
tanθ = (0.56/0.75)
θ = 37° downstream from the starting position

so to land directly across the starting point, the swimmer would need to aim 37° upstream from the starting position. Is this correct? My book says 42° upstream from the starting position, but I can't see how that makes sense.

 

[haruspex]

Please explain your working. How do you get this:

water relative to ground = 0.56 m/s

Actually, you do not need to know any speeds. The two distances are enough information.
(I'm getting 49 degrees.)

 

[gneill]

I wonder if the book is considering the angle to be between the shoreline and the swimmer's trajectory?

 

[haruspex]

I wonder if the book is considering the angle to be between the shoreline and the swimmer's trajectory?

That's probably the explanation, though for that I make it much closer to 41 degrees.

 

[WorkingHard2017]

Hi there, so to get 0.56 m/s, I determined that since the swimmer travels 0.75 m/s, he would travel 72m in 96s. From there, I divided the length of the river, (54m) by 96s to get a speed of 0.56m/s.

 

[WorkingHard2017]

I wonder if the book is considering the angle to be between the shoreline and the swimmer's trajectory?

That's probably the explanation, though for that I make it much closer to 41 degrees.

Could you please explain how you came about that answer?

 

[haruspex]

Hi there, so to get 0.56 m/s, I determined that since the swimmer travels 0.75 m/s, he would travel 72m in 96s. From there, I divided the length of the river, (54m) by 96s to get a speed of 0.56m/s.

Ok. Your error is in assuming that the angle that the swimmer gets carried downstream when swimming straight across relative to the water is the same as the angle the swimmer has to aim upstream. E.g. when aiming upstream the time taken to cross increases, so the current gets to carry the swimmer further.
Draw the right-angled vector triangle for the swimming upstream case. Which side represents the swimmer's velocity relative to the water and which side the current?