Math Problem Statement
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?
Solution
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:
- = Speed of the boats relative to the river (this is the same for both boats).
- = Speed of the river relative to the shore.
- The speed of the first boat relative to the shore is m/s.
- The speed of the second boat relative to the shore is m/s.
(a) Speed of the boats relative to the river :
We have two equations:
Adding these two equations:
So, the speed of the boats relative to the river is 4.5 m/s.
(b) Speed of the river relative to the shore :
Now, subtract the second equation from the first:
So, the speed of the river relative to the shore 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:
- How would the observed speeds change if the river's speed increased?
- 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?
- How would the problem change if the boats were moving in the same direction relative to the river?
- How can this concept be applied to aircraft flying in wind?
- 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.
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Math Problem Analysis
Mathematical Concepts
Relative Velocity
Algebraic Equations
Formulas
v_b + v_r = 5.0 m/s
v_b - v_r = 4.0 m/s
2v_b = 9.0 m/s
2v_r = 1.0 m/s
Theorems
Relative Motion in One Dimension
Suitable Grade Level
Grades 9-11