Two speedboats are travelling at the same speed relative to the water in opposite directions in a moving river. An observer on the riverbank sees the boats moving at 4.0 m/s and 5.0 m/s. (a) What is the speed of the boats relative to the river? (b) How fast is the river moving relative to the shore? 

Let's solve this step by step.

### Given:
- Two speedboats are traveling at the same speed relative to the water.
- They are moving in opposite directions in a moving river.
- An observer on the riverbank sees the speeds of the boats as 4.0 m/s and 5.0 m/s.

### Let:
- \( v_b \) = Speed of the boats relative to the river (this is the same for both boats).
- \( v_r \) = Speed of the river relative to the shore.
- The speed of the first boat relative to the shore is \( v_b + v_r = 5.0 \) m/s.
- The speed of the second boat relative to the shore is \( v_b - v_r = 4.0 \) m/s.

### (a) Speed of the boats relative to the river \( v_b \):
We have two equations:

\[
v_b + v_r = 5.0 \text{ m/s}
\]
\[
v_b - v_r = 4.0 \text{ m/s}
\]

Adding these two equations:

\[
(v_b + v_r) + (v_b - v_r) = 5.0 + 4.0
\]

\[
2v_b = 9.0 \text{ m/s}
\]

\[
v_b = 4.5 \text{ m/s}
\]

So, the speed of the boats relative to the river \( v_b \) is **4.5 m/s**.

### (b) Speed of the river relative to the shore \( v_r \):
Now, subtract the second equation from the first:

\[
(v_b + v_r) - (v_b - v_r) = 5.0 - 4.0
\]

\[
2v_r = 1.0 \text{ m/s}
\]

\[
v_r = 0.5 \text{ m/s}
\]

So, the speed of the river relative to the shore \( v_r \) is **0.5 m/s**.

### Final Answers:
(a) The speed of the boats relative to the river is **4.5 m/s**.

(b) The speed of the river relative to the shore is **0.5 m/s**.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How would the observed speeds change if the river's speed increased?
2. What would be the speed of the boats relative to the river if the observer on the shore saw them both moving at 5 m/s?
3. How would the problem change if the boats were moving in the same direction relative to the river?
4. How can this concept be applied to aircraft flying in wind?
5. What would the observer see if the river's speed matched the boats' speed relative to the river?

**Tip:** When dealing with relative velocities, always consider the direction and combine velocities algebraically (add or subtract) based on the relative motion.

Two speedboats are travelling at the same speed relative to the water in opposite directions in a moving river. An observer on the riverbank sees the boats moving at 4.0 m/s and 5.0 m/s. (a) What is the speed of the boats relative to the river? (b) How fast is the river moving relative to the shore?

Explore how to calculate the speed of two speedboats relative to a moving river and the river's speed relative to the shore. This problem involves concepts of relative velocity and algebraic equations, suitable for students in Grades 9-11.

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