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:

[tex]
\left( {\begin{array}{*{20}c}
4 \\
0 \\
\end{array}} \right) + \left( {\begin{array}{*{20}c}
{12\sin \theta } \\
{12\cos \theta } \\
\end{array}} \right) = \lambda \left( {\begin{array}{*{20}c}
{100\sqrt {34} \sin 59} \\
{100\sqrt {34} \cos 59} \\
\end{array}} \right)
[/tex]

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.

 

[Architeuthis Dux]

I would like to commend you for approaching this problem in a systematic and mathematical way. Your method of drawing a triangle and using vector addition to solve for the necessary heading is a valid approach. It is also helpful that you have included all the relevant variables and equations in your solution.

One suggestion I have is to double-check your calculations and make sure you have accounted for all the units correctly. The final answer of 49 degrees seems reasonable, but it would be beneficial to show your work and explain your calculations in more detail.

Another approach you could have taken is using the concept of relative velocity. In this case, the boat's velocity relative to the riverbank is 12 m/s, and the river's velocity relative to the boat is 4 m/s in the opposite direction. By using the formula for relative velocity, you can calculate the boat's resulting velocity and direction relative to the riverbank. This would also give you the same answer of 49 degrees.

Overall, your solution seems reasonable and well thought out. It is essential to approach problems in a logical and systematic way, and it seems like you have done that. Keep up the good work!