Final answer:
After solving the simultaneous equations based on the observer's recorded speeds of the boats and their known oppositional movement, we find that the speed of the boats relative to the river is 4.5 m/s and the speed of the river relative to the shore is 0.5 m/s.
Step-by-step explanation:
Calculating Speeds of Speedboats and River Current
To solve this problem, let's use the information given that two speedboats are traveling in opposite directions relative to a moving river. Let's denote the speed of the boats relative to the river as Vboat, and the speed of the river relative to the shore as Vriver. Since the observer on the riverbank sees one boat moving at 4.0 m/s and the other at 5.0 m/s, these speeds are the resultant of the boat's speed and the river's current.
Assuming the river flows in the direction in which the boat is seen moving at 5.0 m/s, we can express the speeds observed by the riverbank observer as:
- Speedboat 1 (moving with the current): Vboat + Vriver = 5.0 m/s
- Speedboat 2 (moving against the current): Vboat - Vriver = 4.0 m/s
By solving these two equations simultaneously, we can find that:
- Vboat = (5.0 + 4.0) / 2 = 4.5 m/s
- Vriver = 5.0 - Vboat = 5.0 - 4.5 = 0.5 m/s
Therefore, the speed of the boats relative to the river is 4.5 m/s, and the speed of the river relative to the shore is 0.5 m/s.