Swimming with current, velocity, angles

[Chemfriend]

Hi, I have a homework and I am not sure about some things.

1. Homework Statement

Person A and person B are next to river. Velocity of river is 1 m/s. There is a person drowning 30 m from shore. He doesn´t move, because there is a something, which he can hold. People A and B are 40 far from him (40 m along river, but 50 m if they swum). Person A is a very good swimmer (1,25 m/s) so he decided to jump into the river and swum across the river to person C. Person B can run 4,75 m/s but swims only 1 m/s. So, he runs and then jumps into the river and swims.

a) What angle should person A swim to person C, in order to be in the shortest time to person C?

b) What time would it take?

c) How much time should person B run (and then jump into the river) to get to person C at the same time as person A?

d) How many m has person B already run on the shore?
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It is pictured in image, which i painted: https://s22.postimg.org/y4v78ilkh/whatever.png

Homework Equations

v = d/t
Pytagoras sentence (50^2=30^2+40^2)
cos(-1)(0,8)=36,869°

The Attempt at a Solution

I tried to draw the image in paint.
I counted 50 m with pythagoras sentence.
I can also count the angle of swimmer A (36,87°).
Now, I am not sure how to count velocity of person A. I know velocity of swimmer (1,25 m/s), I know velocity of river (1 m/s). And now to count velocity of swimmer A in river? Should I sum the velocities (1+1,25 = 2,25 m/s) Or is is different because of angle? Now the time will be easy to count (v=d/t). Now I don't know the time, so I can't solve person B.
I enclose image:
https://s22.postimg.org/y4v78ilkh/whatever.png

 

[PeroK]

Hi, I have a homework and I am not sure about some things.

1. Homework Statement

Person A and person B are next to river. Velocity of river is 1 m/s. There is a person drowning 30 m from shore. He doesn´t move, because there is a something, which he can hold. People A and B are 40 far from him (40 m along river, but 50 m if they swum). Person A is a very good swimmer (1,25 m/s) so he decided to jump into the river and swum across the river to person C. Person B can run 4,75 m/s but swims only 1 m/s. So, he runs and then jumps into the river and swims.

a) What angle should person A swim to person C, in order to be in the shortest time to person C?

Think about what happens if A tries to swim directly across the river. What happens to him?

 

[Chemfriend]

Then he is moving with river's current. If he swum directly across the river his velocity would be: v=sqrt((1,25)^2-(1)ˇ2) = 0,75 m/s ? But he is swimming with angle and this is the problem.

 

[PeroK]

Then he is moving with river's current. If he swum directly across the river his velocity would be: v=sqrt((1,25)^2-(1)ˇ2) = 0,75 m/s ? But he is swimming with angle and this is the problem.

That's slower than someone who just floated in the river and was taken at $1m/s$ by the river's current?

 

[Chemfriend]

I think that if he swum with the flow, I would have to sum the velocities (1+1,25=2,25 m/s). If he swum against the flow, it would be 1,25-1=0,25 m/s. And if he swum directly across the river? Now I am not sure, because you are right.

 

[PeroK]

I think that if he swum with the flow, I would have to sum the velocities (1+1,25=2,25 m/s). If he swum against the flow, it would be 1,25-1=0,25 m/s. And if he swum directly across the river? Now I am not sure, because you are right.

Okay. The first step is to work out what direction A moves if he tries to swim across the river. After that, you can work it out for an angle.

Hint: velocity is a vector.

 

[Chemfriend]

https://s12.postimg.org/dh5jgmazx/vector.png
I think this is it.
(v1)*(v2)* sin 90°?

 

[PeroK]

https://s12.postimg.org/dh5jgmazx/vector.png
I think this is it.
(v1)*(v2)* sin 90°?

You may need to revise vectors and vector addition. The swimmer's overall velocity will be the vector sum of the swimmer's velocity (relative to the river) and the river's velocity. In your diagram, you need to put the red and blue vectors end-to-end and the yellow vector should be the result.

This is then also the case where the red vector is at an angle.

These questions are going to be very difficult if you can't work with vectors.

 

[Chemfriend]

I think it should be like this:
https://s11.postimg.org/jeumzswkz/result.png?2

 

[PeroK]

I think it should be like this:
https://s11.postimg.org/jeumzswkz/result.png?2

Okay. Now, you need to draw the swimmer moving at an angle and you have some serious trigonometry to do.

 

[Chemfriend]

https://s16.postimg.org/6m88cigzp/now.png
Is this right?

 

[PeroK]

https://s16.postimg.org/6m88cigzp/now.png
Is this right?

No. You have the river moving at an angle. Second, you can't just guess the angle. Although you could estimate it with a very accurate diagram.

You'll need to know the law of sines and law of cosines for this one.

Given your level of ability with vectors, you may find this a very difficult problem.

 

[Chemfriend]

Now this should be right.
https://s11.postimg.org/4n4mgkifn/thisshouldberight.png
The angle is 36,87 because of pythagoras sentence? (50^2=30^2+40^2) and then cos-1 0,8 = 36,87.

 

[PeroK]

Now this should be right.
https://s11.postimg.org/4n4mgkifn/thisshouldberight.png
The angle is 36,87 because of pythagoras sentence? (50^2=30^2+40^2) and then cos-1 0,8 = 36,87.

That's the angle of the direct line to C, which you can calculate immediately. The question is asking what angle A has to swin relative to the river to achieve this. That's a much harder question.

In any case, you'll need this for part b).

 

[Chemfriend]

So we don't need the angle 36,87? Now it seems much harder as I thought. The only things we need are velocity of swimmer, distance to C (50 m) and velocity of river? SO we need to know what angle he needs to swim to achieve 50 m? Is this the hardest part or the rest is harder? Can you please give me just one equation? or to draw it? Now I have lost the idea.

 

[PeroK]

So we don't need the angle 36,87? Now it seems much harder as I thought. The only things we need are velocity of swimmer, distance to C (50 m) and velocity of river? SO we need to know what angle he needs to swim to achieve 50 m? Is this the hardest part or the rest is harder? Can you please give me just one equation? or to draw it? Now I have lost the idea.

He needs to swim at an angle so that when you add the river's velocity vector to his velocity vector, you get a vector at $36.9°$. If you draw this on a diagram and use the law of sines you'll get the additional angle. The total angle must be greater than $36.9°$, so that the river pushes him onto the real angle of $36.9$.

I'll be offline now.