You have a boat that is capable of moving at 11.8 m/s through still water. You wish to cross a river that flows due south at 5.4 m/s. At what numerical compass heading must you pilot your boat so that you will reach a destination that is due east of your current position?
cos (heading) = 5.4/11.8
Vb - 5.4i = 11.8.
Vb = 11.8 + 5.4i = 13m/s[24.6o] N. of E.
= Compass heading.
9-5.4i) =
To determine the numerical compass heading you need to pilot your boat, we must consider the relative velocities of the boat in still water and the river's flow.
Given:
- Boat's speed in still water: 11.8 m/s
- River's flow speed: 5.4 m/s
To reach a destination that is due east, you need to counteract the effect of the river's flow by angling the boat appropriately. This can be achieved by applying the concept of vector addition.
Step 1: Find the relative velocity of the boat with respect to the river. Since the boat is moving north (due to river's flow) and you want to reach a destination due east, the relative velocity would be at a 90-degree angle to the southward river flow.
Using the Pythagorean theorem, the relative velocity magnitude can be obtained:
relative_velocity = √((boat_velocity)^2 + (river_flow_velocity)^2)
relative_velocity = √((11.8 m/s)^2 + (5.4 m/s)^2)
relative_velocity ≈ √(139.24 + 29.16)
relative_velocity ≈ √168.4
relative_velocity ≈ 12.99 m/s
Step 2: Use trigonometry to calculate the angle required. We'll use the following trigonometric function:
tan(angle) = (opposite/adjacent)
tan(angle) = (river_flow_velocity/boat_velocity)
angle = atan(river_flow_velocity/boat_velocity)
angle = atan(5.4 m/s / 11.8 m/s)
angle ≈ atan(0.4576)
angle ≈ 25.2°
So, to reach a destination that is due east from your current position, you need to pilot your boat at a compass heading of approximately 25.2°.