Boat heading and relative velocity

[drjohn15]

Jerome and Paul are competitive brothers. They live on a small farm on the northern bank of a river that runs purely east and west and that flows to the east at a rate of 1.25 m/s. The brothers have run some time trials on the farm pond, and they know that, in still water, Jerome can paddle the family canoe at a steady rate of 2.9 m/s for a considerable length of time. When Paul runs a considerable distance, it turns out that he can maintain just this same pace.

The brothers like to visit their Uncle Leo who lives on the southern bank of the river. The river is wide at this point, 1410 m across, and their uncle's dock is 170 m to the east of the point which is directly across the river from the brothers' house. Paul is not nearly as strong a paddler as is Jerome, but paddling together they can maintain a paddling speed of 3.48 m/s in the farm pond. Jerome knows that if they point their canoe due south, they will always end up to the east of Uncle Leo's dock by the time they have paddled across the river. He wants to know in which direction they should head to arrive exactly at Uncle Leo's dock without any wasted effort. Paul is finally able to determine the proper direction by using the Law of Sines, which he has learned in his high school geometry class. Make a proper drawing to express the sum of velocities for this problem, and figure out how Paul was able to determine the direction.

Law of Sines : a/sin(A) = b/ sin(B) = c/sin(C) (...or the reciprocal)

Speed when boat heading = South : $\sqrt{3.48^2 + 1.25^2} = 3.698\ m/s$

Direction of Velocity when boat heading = South : $\arcsin \frac{1.25}{3.698} = 19.756\deg\ E\ of\ S$

I'm not sure if I've set up the problem correctly, but I've drawn a right triangle:

start of the boat at the farm,
a vertical velocity vector directly south,
a horizontal velocity vector directly east,
and the addition of those two vectors

and the same triangle but with the distances.

I'm a little lost as to how I can use the Law of Sines to find the proper angle, and honestly I'm really not sure if I've even drawn the scenario correctly.

Any help would be greatly appreciated. Thanks!

 

[Simon Bridge]

If the boat were pointed due south, it will end up east of the dock ... i.e. they will travel too far!
If they want to travel a shrter distance east, which direction should they angle their boat?

It helps to see your reasoning if you (a) show us your diagram, and/or (b) show us your working using variables rather than the actual numbers.

i.e. the river is w wide and flows with speed c due east.
Jerome and Paul together paddle at speed v wrt the water.
By himself, Jerome paddles at speed u wrt the water, and Paul runs at this speed.
The uncles dock is a distance d down the bank.

If you put the origin on the brother's house, with the y-axis pointing north, then the uncles house is at position (-w,d). See how this makes things clearer?. The brothers want to paddle directly at this point from the origin. What total velocity do they need to have?

 

[drjohn15]

Thank you for your response!

I realized that the triangles that I had drawn were correct, I just didn't realize that I was using the wrong angle.

 

[Simon Bridge]

Well done :)

 

[Nickie]

Great job setting up the problem! You have correctly drawn the right triangle with a vertical velocity vector pointing south and a horizontal velocity vector pointing east. The distance across the river, 1410 m, is the hypotenuse of the triangle, and the distance from the starting point to Uncle Leo's dock, 170 m, is the horizontal leg of the triangle.

To use the Law of Sines, we need to find the angle opposite the 170 m side. We can do this by finding the angle opposite the 1410 m side and subtracting it from 90 degrees. The angle opposite the 1410 m side can be found using the inverse sine function, as you have correctly done. So, the angle opposite the 170 m side is 90 degrees minus 19.756 degrees, which is 70.244 degrees. This is the direction that the brothers should head in order to arrive exactly at Uncle Leo's dock without any wasted effort.

It's great that Paul was able to use his knowledge of the Law of Sines to determine the proper direction. This is a useful application of geometry in real life situations. Keep up the good work!