Relative Velocity: Swimming Angle to Reach Shore

[mikeh3593]

Homework Statement

Deloris is being chased through the forest by hungry cannibals. He decides to evade the flesh-eating spider monkeys by crossing a river. He Arrives at the bank of the river 10m[N] of a safe landing on the opposite side. the river is moving at a speed of 3.5 m/s . if he has a maximum swimming speed of 3 m/s, what angle must he swim at in order to make it to the opposite shore?

Homework Equations

V(deloris to the shore) = V(Deloris to the water) + V(Water to the Shore)

The Attempt at a Solution

So far I have used the pythagorean theorem to get the speed 4.61m/s
also to get the distance traveled to the end point to be 22.36m
and the angle of displacement i obtained using tangent which is 63.4 degrees

 

[JesseC]

I think it would help you and us to draw a picture! How wide is the river?

 

[mikeh3593]

Oh yeah! forgot, sorry, the river is 30m wide.

| |
| |
X | up is [N]
| |
| 10m |
| |
|---------X
| 30m |
| |
| |

sorry I am new i don't know how to make a picture

 

[JesseC]

I think this is what you mean?

[PLAIN]http://img213.imageshack.us/img213/8501/riverz.jpg

Basically you're trying to find the direction he should swim so that he ends up traveling at an angle x from the south direction... (along the blue arrow) I hope my diagram is clear.

The first thing to do would be to work out the angle x.

I have suspicion that he will never make it to the other side :) ? Is this allowed as an answer?

If you choose a nice set of co-ordinates you can find an answer by solving one equation...

 

[mikeh3593]

how would i go about this? I've been stressing about this question and now i can't even wrap my head around it i have the angle which is 63.4 degrees.

 

[JesseC]

To calculate x, i get:

$$tan(x)= \frac{30}{10}$$
$$x = 71.6$$ degrees

EDIT: x was a bad choice of letter for this angle... from now on let's just call it a.

 

[mikeh3593]

Oh my bad the distance was 20m sorry so what where do I go after from getting the angle for a?

 

[JesseC]

I hope you know about resolving vectors to their components!

[PLAIN]http://img593.imageshack.us/img593/5049/river.jpg

If you put a co-ordinate axis onto your problem as i drew in the picture, you can hopefully see: that in order for him to travel in the x direction (towards the safe shore) he better not have any velocity in the y direction.

So we don't know what angle he is going to swim at, but let's call it b. How can he not have any velocity in the y direction? Only if his y direction of velocity is equal and opposite to that of the rivers.

Mathematically you write this:

$$v_{river} sin(a) = v_{deloris} sin(b)$$

If you know velocity of the river, you know a and you know velocity of Deloris then you can find b. What do you get?

 

[JesseC]

Ok just do everything I said except use the right distance for the river...

It would be impossible for him to get past if the river was 30m! 20m, he can make it :D

EDIT: Oh... maybe not :P, sure he can't swim any faster? haha

 

[mikeh3593]

Okay! I have got 73.46 degrees as the final answer, I understand this now! Thank you so much!