Boat on River - Relative velocities in two dimensions

Description : Let's get IceCream!! You have a boat that has a fixed speed relative to the water, and you want to get across the river. The simulation initially shows the boat aiming straight across the river (and ending up downstream). There is a dock (in grey) directly across the river .. it would be nice to get there. But there is an IceCream place just upriver from the dock, it would be BETTER to get there! See how many questions you can answer below before changing the angle (hint - measure distance across, and distance downriver, and time - that should allow you to calculate all velocities!).
   [Until I can figure out how to angle the boat in the animation, we will use a red ball and an arrow to indicate the boat direction.] The red arrow is the direction of the velocity vector of the boat with respect to (w.r.t.) the water and shows the direction that the boat is pointing . The blue arrow shows the velocity of the Water w.r.t. the Shore. The black arrow (shown when the box is checked) is the velocity of the boat w.r.t. the Shore.

Questions as you run the initial animation: (Click the Play button first)
A) What is the speed of the boat relative to still water?
B) How fast is the river going sideways?
C) What angle is needed upstream to exactly move across?

Now we want to aim the boat upstream slightly .. in order to have it go exactly across to the gray Dock. Change the angle we must tilt the boat upstream in the box below, and press the Update Angle button, and then run the simulation ... try your calculation for the correct angle to have the boat go exactly across. Don't click in the check box to show the vector addition until you play the animation and try to calculate the angle from theory. How'd you do? The calculation for the correct angle to get to the dock can be done with our standard "vector velocity" equations. Make sure you can do it! [Note: the velocity of the Water w.r.t. the Shore and velocity of the Boat w.r.t. the Water are the same for all of these angles.

           degrees upstream that you point the boat (red arrow)    

When Checked, Shows the head to tail addition of the velovity vectors .

Enough of this .. let's get IceCream! Find the angle that will let the boat move directly toward the IceCream shack. (The spatial angle to the icecream shack is 36.6 degrees - is that the best angle? Why not?) What angle is needed to exactly arrive at the icecream place? What is the quickest you can cross the stream? How much extra time to go exactly to the dock, to the icecream place?

Note: The calculation to pre-calculate the correct ice cream angle is challenging (beyond our standard effort) .. can you do it? (Hint, think of what physical direction we want to move the boat .. then play with the laws of sines and cosines.) It is easier to use trial and error to get the right angle and then verify the angle with the correct equations than solve the equations for the angle that the boat's vector makes.

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