# Homework Help: Relative Velocity-Boat Problem and Minimization

1. Sep 8, 2011

### gbean

1. The problem statement, all variables and given/known data
Fred's friends are in a boat. If they could travel perpendicularly to the shore, they could land at his position. However, a strong current vc is greater than the maximum vm of the motor. Find the magnitude of the angle, measured relative to the straight-across direction, at which his friends should point the boat to minimize the distance Fred has to walk.

a) arcsin (vm/vc)
b) arctan (vc/vm)
c) tan (vc/vm)
d) arctan (sqrt(vc/vm))

2. Relevant equations
a^2 + b^2 = c^2
sin(theta) = opposite/hypotenuse (=> theta = arcsin(opposite/hypotenuse))
vx = vcos(theta)
vy = vsin(theta)
v = vx+vy

3. The attempt at a solution
I tried to draw this out and I get that the angle is represented by theta = arctan(vm/vc).

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Last edited: Sep 8, 2011
2. Sep 8, 2011

### gbean

I know that the answer is A, but I still keep getting arcctan...

3. Sep 8, 2011

### diazona

It's not clear from the diagram that you understand what $\vec{v}_c$ and $\vec{v}_m$ are. $\vec{v}_c$ is the velocity of the current (relative to the shore), and $\vec{v}_m$ is the velocity of the boat (hint: relative to what?).

4. Sep 8, 2011

### gbean

Relative to an observer on the shore, aka Fred?

5. Sep 8, 2011

### gbean

Vm is the velocity perpendicular to the shore, and then Vc is the velocity parallel to the shore?

6. Sep 8, 2011

### diazona

Nope. Think about this: $v_m$ represents the speed at which the motor is able to push the boat. What is the motor pushing against? Therefore, what should the velocity $\vec{v}_m$ be measured against?

7. Sep 8, 2011

### gbean

The motor is pushing against vc, the current?

8. Sep 8, 2011

### diazona

Well... sure, you could say that. So what is $\vec{v}_m$ relative to?

9. Sep 8, 2011

### gbean

To the current? I'm not totally sure what I'm supposed to be gleaning.

10. Sep 8, 2011

### diazona

Yep.
So how could you express the velocity of the boat relative to the shore?

11. Sep 8, 2011

### gbean

Vo = Vm + Vc

Vo = velocity relative to the shore
Vm = velocity of boat
Vc = velocity of current

12. Sep 8, 2011

### diazona

OK, good. (I took a few minutes to work through it to make sure I wasn't leading you down the wrong track)

Now, the velocity of the current $\vec{v}_c$ is fixed, but the velocity of the boat relative to the current, $\vec{v}_m$, can be pointed in any direction. So I would suggest drawing a new diagram of the river. Include the vector $\vec{v}_c$ and a circle representing all the possibilities for $\vec{v}_m$.

13. Sep 8, 2011

### gbean

Like this?

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14. Sep 8, 2011

### diazona

Kind of like that, but put the tail of $\vec{v}_m$ (the center of the circle) at the tip of $\vec{v}_c$, because you're adding them.

15. Sep 8, 2011

### gbean

So like this?

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16. Sep 8, 2011

### diazona

Sure, I guess that'll work. Now which orientation of $\vec{v}_m$ satisfies the conditions of the problem?

17. Sep 8, 2011

### gbean

Vm pointing to the left, or upstream.

18. Sep 8, 2011

### diazona

No; if $\vec{v}_m$ points directly upstream, parallel to $\vec{v}_c$, the boat will never reach the shore. Do you understand why?

Here's the way to think about it: when you add two vectors using the tail-to-tip method, the sum is a vector pointing from the tail of the first one ($\vec{v}_c$) to the tip of the second one ($\vec{v}_m$). The tip of the second vector, in this case, is on the circumference of the circle. So the sum $\vec{v}_0$ (that's what you called it, right?) should be drawn from the tail of $\vec{v}_c$ to a point on the circle. Of all the possible vectors you can draw this way, which one gets the boat to the shore closest to Fred?

19. Sep 8, 2011

### gbean

This is what I intuitively want vm to point, but I can't explain it...

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20. Sep 9, 2011

### NewtonianAlch

Hmm, you said you know the answer is A right? Have you considered drawing that situation to fit that model and seeing why it is so?

Vm should be the vector going towards Fred on the other side but at an angle, and Vc should be the vector going parallel to him, i.e. upstream/downstream.

Therefore to satisfy the triangle, one more side needs to be drawn, and that is the resultant vector that hopefully gets the boat straight to him.