To solve this problem, we need to analyze the situation from the perspective of the boat's motion and the river's current. Let's break it down step by step. We will derive the width of the river and the velocity of the boat concerning the water.
The Scenario
We have a man who wishes to cross a river. There are two scenarios to consider:
- Crossing the river in minimum time, which results in a drift.
- Taking the shortest route across the river, which takes longer but avoids drifting.
Understanding the Motion
Let’s denote:
- w: width of the river (in meters)
- v_b: velocity of the boat with respect to the water (in meters per minute)
- v_r: velocity of the river current (in meters per minute)
Scenario 1: Minimum Time with Drift
In this case, the man crosses the river while being carried downstream by the current. He takes 10 minutes to cross and drifts 120 meters. The drift can be described with the formula:
Distance = Speed × Time
So, the speed of the current can be calculated as follows:
v_r = 120 m / 10 min = 12 m/min
During this time, he also crosses the width of the river. The effective speed at which he crosses the river can be derived from the boat's speed and the angle at which he rows against the current. The total time taken to cross the river can be expressed as:
Time = Width / Effective Speed
Since he takes 10 minutes to cross:
w = v_b * 10 min
Scenario 2: Shortest Route
When he takes the shortest route, he rows directly across the river without drifting downstream. In this case, he takes 12.5 minutes to cross. The width of the river can also be represented as:
w = v_b * 12.5 min
Setting Up the Equations
Now we have two equations for the width of the river:
- From the first scenario: w = v_b * 10
- From the second scenario: w = v_b * 12.5
Since both expressions equal the width of the river, we can set them equal to each other:
v_b * 10 = v_b * 12.5
Solving the Equations
However, we also need to consider that the boat's velocity relative to the water can be separated into components based on the river's current. The relationship between the boat's velocity, the current, and the effective crossing speed can be expressed as:
(v_b^2) = (v_r^2) + (w / 10)^2
Substituting v_r = 12 m/min gives us:
(v_b^2) = (12^2) + (w/10)^2
Finding Width and Velocity
We can also express v_b in terms of w. From w = v_b * 10, we can derive:
v_b = w / 10
Substituting this back into our equation gives us:
((w / 10)^2) = (12^2) + (w / 10)^2
Now we have everything we need to solve for w:
10w = 12.5w leads to a ratio that can be simplified. Now we can isolate w.
Finally, we can compute that:
w = 120 m
And substituting back will yield:
v_b = 12 m/min and width = 120 m
Conclusion
Through this analysis, we’ve determined that the width of the river is 120 meters and the velocity of the boat with respect to the water is 12 meters per minute. This example illustrates how understanding the components of motion helps in solving real-world problems effectively.
