To determine the time it would take for the boat to cross the river in the shortest path, we first need to analyze the situation presented. The boat is crossing a river that is 200 meters wide and experiences a drift of 100 meters due to the current. This means that while the boat is trying to go straight across, the current is pushing it downstream.
Understanding the Current's Effect
When the boat crosses the river, it has two components of motion: one is the motion across the river (perpendicular to the current), and the other is the motion downstream (parallel to the current). The drift of 100 meters indicates how far downstream the boat has been pushed by the current by the time it reaches the opposite bank.
Calculating the Time Taken
Let’s denote the width of the river as w = 200 m and the drift as d = 100 m. The time taken to cross the river in the scenario described is t. The boat's speed across the river can be calculated using the formula:
In this case, the distance is the width of the river (200 m), and the time taken is t. Therefore, the speed of the boat across the river is:
- Speed across the river = 200 / t
Finding the Speed of the Current
Now, we need to consider the drift caused by the current. The drift of 100 meters occurs while the boat is crossing the river. The time taken to cross the river is the same time during which the current is pushing the boat downstream. Thus, the speed of the current can be calculated as:
- Speed of current = Drift / Time = 100 / t
Crossing the River in the Shortest Path
To cross the river in the shortest path, the boat must aim upstream at an angle that compensates for the current. This means that the boat's velocity must have a component that counters the current's speed. The effective speed of the boat can be calculated using the Pythagorean theorem, as the boat's speed across the river and the current's speed form a right triangle.
Calculating the Effective Speed
The effective speed of the boat when crossing straight across can be expressed as:
- Effective speed = √((Speed across the river)² + (Speed of current)²)
Substituting the values we have:
- Effective speed = √((200/t)² + (100/t)²)
- Effective speed = √((40000/t²) + (10000/t²))
- Effective speed = √(50000/t²) = 100√5/t
Calculating the Time for the Shortest Path
Now, to find the time taken to cross the river in the shortest path, we can use the formula:
Substituting the width of the river and the effective speed:
- Time = 200 / (100√5/t)
- Time = (200t) / (100√5)
- Time = (2t) / √5
Thus, the time taken to cross the river by the shortest path will be (2t) / √5.
Summary
In summary, the time taken for the boat to cross the river in the shortest path is (2t) / √5, where t is the time taken in the initial scenario where the boat drifts downstream. This approach ensures that the boat compensates for the current, allowing it to reach the opposite bank directly across from its starting point.
