Solving a Boat Navigation Problem: A Family's Journey Downstream

[danago]

Homework Statement

A Family has a boat which can travel at 12 m/s in still water. They are using it to reach a point 500m downstream on the other side of a river 300m wide, and flowing at 4 m/s. What heading must they take, relative to their starting point, in order to reach their destination?

The Attempt at a Solution

I first drew a triangle, with the side parallel to the river banks 500m, and the side perperndicular 300m. From there, i found that to end up 500m downstream, the resultant direction of travel should be ~59 degrees from the perpendicular (or 31 degrees from the bank).

The rivers current plus the boats motion should result in a velocity in the direction of 59 degrees. So:

$$\left( {\begin{array}{*{20}c}<br /> 4 \\<br /> 0 \\<br /> \end{array}} \right) + \left( {\begin{array}{*{20}c}<br /> {12\sin \theta } \\<br /> {12\cos \theta } \\<br /> \end{array}} \right) = \lambda \left( {\begin{array}{*{20}c}<br /> {100\sqrt {34} \sin 59} \\<br /> {100\sqrt {34} \cos 59} \\<br /> \end{array}} \right)$$

I solved for theta, and found that it equals 49 degrees. So i concluded that to end up 500m downstream, the boat should set off at 49 degrees to the perpendicular of the banks.



Im unsure if that is correct. Also, is there a better way i should have gone about it?

Thanks,
Dan.

 

[Gib Z]

Are the matrices really necessary? Other than that, that seems like a good way to do it. You can check if its correct by yourself, add up the vectors assuming 49 degrees is correct, see if it gets you where you want.