A ferry boat operator is required to drive directly to the opposite side of the river (starting at south side of river is A and the north side of river is B where boat will end up) but does not know where to point the nose of the boat. The river is flowing due West at 4 mph and the boat can bravel 9 mph over still water. What direction (CW relative to N) should the ferry driver point the nose of the boat?
Draw a diagram. Draw your line straight north from A, and a line east from A of length 4. The hypotenuse is the speed of the boat, 9.
That is, for every 9 miles the boat travels in the desired direction, it will drift 4 miles west. ending up directly north of A.
So, the angle a, reckoning from the north direction is such that
sin(a) = 4/9 = .4444
so, a = N 26.4° E
To solve this problem, we need to consider the relative velocities of the boat and the river.
Let's break down the velocities involved:
1. The boat's velocity relative to the river: Since the boat travels at 9 mph over still water, this is the speed at which it moves without any influence from the river.
2. The river's velocity: The river is flowing due West at 4 mph. This means that the velocity of the water is 4 mph in the West direction.
3. The resultant velocity: This is the velocity at which the boat will actually move, considering the combined effect of its velocity and the river's velocity.
To determine the resultant velocity, we'll use vector addition:
1. Start by drawing a diagram with North at the top and West to the left.
2. Label the boat's velocity vector as 9 mph to the North (straight up).
3. Label the river's velocity vector as 4 mph to the West (straight left).
4. Draw lines from the tail of each vector so they meet at a common point.
5. The direction of the resultant vector (the direction the boat should point) will be the line connecting the tail of the boat's velocity vector to the head of the river's velocity vector.
Using basic trigonometry, we can find the direction of the resultant vector:
1. Calculate the angle the resultant vector makes with the North direction.
- This can be done using the tangent function: tan(theta) = Opposite / Adjacent.
- In this case, the opposite side length is the North component (9 mph) and the adjacent side length is the West component (4 mph).
- So, tan(theta) = 9 mph / 4 mph.
- Solve for theta: theta = arctan(9/4).
2. Convert the angle theta to a clockwise direction relative to North.
- Since the angle we found is the counter-clockwise angle from North, we subtract it from 90 degrees to get the clockwise direction relative to North.
- Therefore, the direction the ferry driver should point the nose of the boat is 90 degrees - arctan(9/4) clockwise from North.
To summarize, the ferry driver should point the nose of the boat approximately 59.04 degrees clockwise from North.