Swimmer's velocity relative to the shore (vectors)

[ulfy01]

Homework Statement

A swimmer is training in a river. The current flows at 1.33 metres per second and the swimmer's speed is 2.86 metres per second relative to the water. What is the swimmer's speed relative to the shore when swimming upstream? What about downstream?

Homework Equations

Pythagoras.

The Attempt at a Solution

Here's my problem. Because we're looking at vectors, I would normally do the vector sum of both the velocities and use Pythagoras, as both given vectors are perpendicular.


V_current = 1.33 m/s
V_{swimmer relative to water} = 2.86 m/s

So the resultant vector would be: $\sqrt{}$(1.33² + 2.86²)

Giving 3.15 m/s, however this is wrong, as the answer given is 1.53 m/s upstream.

I'm puzzled as to how this answer was reached.

 

[azizlwl]

The swimmer is not swimming across the river.
He is swimming against the current. That how normally swimmers trainned.
Now imagine you running in direction to the east at a speed 1.33 metres per second on a train with 1.33 metres per second speed heading west.
To the man on the platform seeing you not moving, but with respect of the train you are running at 1.33 metres per second in easterly direction.

 

[TSny]

Hi, ulfy01.

Note that "swimming upstream" means swimming in a direction opposite to the current, not perpendicular to the current. So, there is no right triangle here.

[oops: I'm a bit late here, sorry.]

 

[ulfy01]

I just realized this and made a fool of myself, really. Way to overthink a problem and not read it properly. I'll go hide in a corner now. Thanks azizlwl!

 

[Nickie]

I would approach this problem by first defining the direction of the current and the swimmer's velocity relative to the shore. Let's say the current is flowing downstream and the swimmer is swimming upstream. In this case, the swimmer's velocity relative to the shore would be the difference between their velocity relative to the water (2.86 m/s) and the velocity of the current (1.33 m/s).

This can be visualized as the swimmer trying to swim against the current, so their actual speed relative to the shore would be slower than their speed relative to the water. Using the Pythagorean theorem to find the resultant velocity would not be correct in this situation.

To find the swimmer's speed relative to the shore when swimming upstream, we can use the following equation:

Vs = Vw - Vc

Where:
Vs = swimmer's velocity relative to the shore
Vw = swimmer's velocity relative to the water
Vc = velocity of the current

Plugging in the given values, we get:

Vs = 2.86 m/s - 1.33 m/s = 1.53 m/s

This is the correct answer given in the problem.

For swimming downstream, the swimmer's velocity relative to the shore would be the sum of their velocity relative to the water and the velocity of the current:

Vs = Vw + Vc = 2.86 m/s + 1.33 m/s = 4.19 m/s

So the swimmer's speed relative to the shore when swimming downstream would be 4.19 m/s. This approach takes into account the direction of the current and the swimmer's velocity relative to the shore, and gives us the correct answer for both upstream and downstream swimming.