Optimizing Boat Direction in a River with Current: A Vector Diagram Approach

In summary, the boat should point slightly upstream with a speed of 10 m/s and the river has a current of 5 m/s. By drawing a right triangle with the boat's speed as the hypotenuse and the river's current as one of the legs, the remaining side represents the boat's speed across the river. Using Pythagoras' theorem, the magnitude of this speed can be calculated to be approximately 8.66 m/s. The angle can then be determined using a trigonometric function, where the opposite side is the river's current and the hypotenuse is the boat's speed.
  • #1
Phyzix
10
0
I know i need to use a simple vector diagram ...but I'm not sure how to set it up...

In what direction should a boat point if it has a speed of 10 m/s, and wants to end up directly across a river with a current of 5 m/s.

help? :confused: :bugeye: :rolleyes:
 
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  • #2
Draw a pictur so that the vector representing your boats speed is pointing slightly up stream (don't worry about the angle now: just label it θ) and has length 10. Draw a horizontal line (across the stream) representing the "course you want to make good" and finally draw a vertical line representing the flow of the stream of length 5.

You should now have a right triangle with angle θ (which you don't know yet), hypotenuse of length 10 and leg opposite θ of length 5.

Do you remember a trig function that is "opposite over hypotenuse"? For what angle is that function equal to 5/10= 1/2?
 
  • #3
You just have to draw a right angled triangle. The length of the long side represents the boat's speed, and one of the shorter sides is the river speed. The length of the remaining side gives the boat's speed across the river.

By Pythagoras:

[tex]5^2 + x^2 = 10^2[/tex]

[tex]x = \sqrt{10^2 - 5^2} \approx 8.66[/tex]

[tex]sin(a) = \frac{5}{10}[/tex]
 
  • #4
ahh thanks! I was thinking along these lines but I couldn't decide if the hypotenuse was the 10 or one of the legs..thanks a lot guys :) :biggrin:
 

What is the "River Boat simple problem"?

The "River Boat simple problem" is a classic mathematical puzzle that challenges individuals to transport a group of objects from one side of a river to the other using a limited number of boats.

What are the rules of the "River Boat simple problem"?

The rules of the "River Boat simple problem" are as follows:

  • The puzzle is set on a river with two banks - a starting bank and a destination bank.
  • There are a set number of objects that need to be transported from the starting bank to the destination bank.
  • Only one or two objects can be transported on a boat at a time.
  • If there are more objects on one bank than the other, the objects cannot outnumber the people on the bank or else they may riot.
  • The objective is to transport all the objects from the starting bank to the destination bank using the minimum number of boat trips.

What is the strategy for solving the "River Boat simple problem"?

The most common strategy for solving the "River Boat simple problem" is to first determine the minimum number of boat trips required by dividing the total number of objects by the maximum number of objects that can be transported on a boat at a time. Then, the objects should be divided into groups that can be transported together without violating the rules.

What is the significance of the "River Boat simple problem"?

The "River Boat simple problem" is often used as an example in computer science and mathematics courses to teach problem-solving techniques and algorithmic thinking. It also serves as a fun and challenging brain teaser for individuals of all ages.

Are there variations of the "River Boat simple problem"?

Yes, there are many variations of the "River Boat simple problem" that involve different constraints and objectives. Some variations may include different numbers of objects, boats with different capacities, and additional rules or conditions. These variations can make the problem more complex and require different strategies for solving.

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