Optimizing Boat Angle for Hitting a Target Across a River

[KurtWagner]

So I have a boat going across a river (y-direction, north) 400m wide.
I am trying to hit a target 75m in the positive x direction on the opposite side of the river.
My boat will travel a velocity v.
The river will provide a constant velocity of +0.5 (east)

I am trying to find the angle a, north of west to point my boat.

hope that all makes sense.

so far I have:

y = vt or t = y/v
t = 400 / v*sin(a)

x = vt
75 = [0.5 - cos(a)] * t

then I plugged in t

75 = 200/[v*sin(a)] - [400*v*cos(a)]/[v*sin(a)]

I can't seem to solve this. Am I doing this the right way?

I was thinking I need another function and I remember having to use
sin(a)^2 + cos(b)^2 = 1
for something like this a while back. And if I do, how do I? Do I just solve the first two equations for sin and cos and then plug the equations into the third, expand them, then solve another for t and then plug it in and simplify?

any help would be greatly appreciated.

 

[BvU]

Hello Kurt, sorry you missed the template. Please use it.

I wonder if your river velocity is 0.5 nautical miles per fortnight or 0.5 m/s, because you missed giving the units. With an error as a consequence:

If you had written your second relevant formula (under 2. in the template) as

75 m = [0.5 m/s - cos(a)] * t

you would have immediately seen that it is nonsense to add a cosine to a velocity (how long is a year plus one ?)

However, this appears to be a typo omission, because under "3. attempt at solution", the v pops up again.

A second error pops up if you temporarily assume the river not to flow. a can then be zero, but instead of t = 400 m / v you get a divergence. You want to interchange cos and sin.

For the rest, you're okay:
vt cos(a) = 400 m plugged into the second eqn gives you
$${75\over 400}v\cos\alpha = 0.5 - v \sin\alpha$$
which is a simple goniometric equation of the form $a \cos\alpha + b\sin\alpha = c$.
The relevant expression to help you solve that is $\sin(\alpha+\beta) = ...$

 

[KurtWagner]

thanks. sorry for missing the units and for the typo also. I'll give it another crack. :)