Boat crossing river problem using reference frames

[leroyjenkens]

Homework Statement

A 110-m-wide river flows due east at a uniform speed of 3.3 m/s. A boat with a speed of 8.6 m/s relative to the water leaves the south bank pointed in a direction 37 degrees west of north. What is the (a) magnitude and (b) direction of the boat's velocity relative to the ground? Give the direction as the angle of the velocity from due north, positive if to the east and negative if to the west. (c) How long does it take for the boat to cross the river?

Homework Equations

I was using the PA = PB + BA equation. Which is supposed to be set up like this: the velocity of something with respect to something else, equals, the velocity of something with respect to something else, plus, the velocity of something with respect to something else. Now I know how to do this problem, because it's simple enough to do without using the equation I just mentioned. But I was supposed to use this equation to do the problem.
What I did instead was I just thought about it logically. The river is moving east and the boat is moving in the westward direction, so I separated the boat's velocity into X and Y components and subtracted the X component from the velocity of the river and then solved from there.

But as far as the equation goes, how do I set up the equation? Which velocity goes on the left side of the equation and which velocities go on the right side? As far as I can tell, it makes a huge difference.
I tried using that equation to solve the problem, but the way I set up the equation, I ended up with an equation that would add the boat's velocity to the river's velocity, which would mean the boat would go faster in the west direction, even though the river is moving east. That would make no sense.
So anyone know how to set up that equation? In my book it's under the "Relative Motion in One direction" section, if that helps.

Thanks.

 

[Nate810]

The Attempt at a SolutionI was using the PA = PB + BA equation. Which is supposed to be set up like this: the velocity of something with respect to something else, equals, the velocity of something with respect to something else, plus, the velocity of something with respect to something else. For this problem, the left side of the equation would be the boat's velocity relative to the ground, and the right side of the equation would be the sum of the boat's velocity relative to the water (8.6 m/s) and the velocity of the water relative to the ground (3.3 m/s). So the equation would look like this: Vboat (relative to ground) = Vboat (relative to water) + Vwater (relative to ground)Using this equation, we can solve for Vboat (relative to ground) by plugging in the known values: Vboat (relative to ground) = 8.6 m/s + 3.3 m/s Vboat (relative to ground) = 11.9 m/s Now that we have the magnitude of the boat's velocity relative to the ground, we can use the angle given in the problem (37 degrees west of north) to determine the direction. So the direction of the boat's velocity relative to the ground is 37 degrees west of north.