Relative Velocity-Boat Problem and Minimization

In summary: We know the angle between vm and vc must be less than 90 degrees, because if it were greater, the boat would never reach the shore. So the angle between vm and the shore must be less than 90 degrees as well, since it is smaller than the angle between vm and vc. Therefore, the angle between vm and the shore must be arcsin(vm/vc).
  • #1
gbean
43
0

Homework Statement


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))

Homework Equations


a^2 + b^2 = c^2
sin(theta) = opposite/hypotenuse (=> theta = arcsin(opposite/hypotenuse))
cos(theta) = adjacent/hypotenuse
tan(theta) = opposite/adjacent
vx = vcos(theta)
vy = vsin(theta)
v = vx+vy

The Attempt at a Solution


I tried to draw this out and I get that the angle is represented by theta = arctan(vm/vc).

I don't know what I'm doing wrong, please help!
 

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  • #2
I know that the answer is A, but I still keep getting arcctan...
 
  • #3
It's not clear from the diagram that you understand what [itex]\vec{v}_c[/itex] and [itex]\vec{v}_m[/itex] are. [itex]\vec{v}_c[/itex] is the velocity of the current (relative to the shore), and [itex]\vec{v}_m[/itex] is the velocity of the boat (hint: relative to what?).
 
  • #4
Relative to an observer on the shore, aka Fred?
 
  • #5
Vm is the velocity perpendicular to the shore, and then Vc is the velocity parallel to the shore?
 
  • #6
gbean said:
Relative to an observer on the shore, aka Fred?
Nope. Think about this: [itex]v_m[/itex] represents the speed at which the motor is able to push the boat. What is the motor pushing against? Therefore, what should the velocity [itex]\vec{v}_m[/itex] be measured against?
 
  • #7
The motor is pushing against vc, the current?
 
  • #8
Well... sure, you could say that. So what is [itex]\vec{v}_m[/itex] relative to?
 
  • #9
To the current? I'm not totally sure what I'm supposed to be gleaning.
 
  • #10
gbean said:
To the current?
Yep.
gbean said:
I'm not totally sure what I'm supposed to be gleaning.
So how could you express the velocity of the boat relative to the shore?
 
  • #11
Vo = Vm + Vc

Vo = velocity relative to the shore
Vm = velocity of boat
Vc = velocity of current
 
  • #12
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 [itex]\vec{v}_c[/itex] is fixed, but the velocity of the boat relative to the current, [itex]\vec{v}_m[/itex], can be pointed in any direction. So I would suggest drawing a new diagram of the river. Include the vector [itex]\vec{v}_c[/itex] and a circle representing all the possibilities for [itex]\vec{v}_m[/itex].
 
  • #13
Like this?
 

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  • #14
Kind of like that, but put the tail of [itex]\vec{v}_m[/itex] (the center of the circle) at the tip of [itex]\vec{v}_c[/itex], because you're adding them.
 
  • #15
So like this?
 

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  • #16
Sure, I guess that'll work. Now which orientation of [itex]\vec{v}_m[/itex] satisfies the conditions of the problem?
 
  • #17
Vm pointing to the left, or upstream.
 
  • #18
No; if [itex]\vec{v}_m[/itex] points directly upstream, parallel to [itex]\vec{v}_c[/itex], 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 ([itex]\vec{v}_c[/itex]) to the tip of the second one ([itex]\vec{v}_m[/itex]). The tip of the second vector, in this case, is on the circumference of the circle. So the sum [itex]\vec{v}_0[/itex] (that's what you called it, right?) should be drawn from the tail of [itex]\vec{v}_c[/itex] 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
This is what I intuitively want vm to point, but I can't explain it...
 

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  • #20
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.
 
  • #21
gbean said:
This is what I intuitively want vm to point, but I can't explain it...
Yes, that is exactly the right configuration of vectors. Now you have a triangle; what can you conclude about any of the angles of that triangle?
 
  • #22
45-45-90, but why?
 
  • #23
gbean said:
45-45-90, but why?
How would I know? You're the one who came up with those numbers :wink:

Seriously though, that's not correct. Here's a hint: you can only know the numeric value of one of the angles in the triangle. Which one, and what is it? (And how do you know?)
 

Related to Relative Velocity-Boat Problem and Minimization

1. What is relative velocity in the context of a boat problem?

Relative velocity refers to the speed and direction of an object with respect to another object. In the context of a boat problem, it is the velocity of the boat relative to the water or the velocity of one boat relative to another boat.

2. How do you solve a relative velocity-boat problem?

To solve a relative velocity-boat problem, you can use the concept of vector addition. First, draw a diagram to represent the situation and label the given velocities. Then, use the vector addition formula to find the resultant velocity. Finally, use the Pythagorean theorem and trigonometric functions to find the magnitude and direction of the resultant velocity.

3. What is the minimization principle in relative velocity-boat problems?

The minimization principle states that the shortest distance between two objects moving at constant velocities is the path that minimizes the distance between them. In the context of relative velocity-boat problems, this means that the boats will take the shortest path to reach their destinations while accounting for their velocities and directions.

4. Can relative velocity-boat problems be solved using the laws of motion?

Yes, relative velocity-boat problems can be solved using the laws of motion, specifically the law of inertia and the law of action and reaction. These laws help us understand how objects move in relation to one another and can be used to solve for velocities and directions in relative velocity-boat problems.

5. Are there any real-life applications of relative velocity-boat problems?

Yes, relative velocity-boat problems have many real-life applications, such as in sailing and navigation. The concept of relative velocity is also used in other fields, such as aviation and space travel, to calculate trajectories and flight paths. Additionally, understanding relative velocity is important for predicting and avoiding collisions between moving objects.

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