To solve this problem, we need to analyze the situation involving the boat's movement in a river and the wind's effect on its direction. The boat's velocity in still water and the river's current will help us determine the wind's components. Let's break it down step by step.
Understanding the Scenario
The river flows eastward at a velocity of Vr = 1 m/s, and the boat has a velocity in still water of Vb = 3 m/s. When the boat travels downstream, it points northwest, and when it travels upstream, it points northeast. This indicates that the wind is affecting the boat's direction.
Analyzing Downstream Movement
When the boat moves downstream, its effective velocity is the sum of its velocity in still water and the river's current. The boat's velocity vector can be represented as:
- Downstream velocity: Vb + Vr = 3 m/s + 1 m/s = 4 m/s (eastward)
Since the flag points northwest, we can deduce that the wind has a component pushing the boat towards the northwest. This means the wind has both a northward and an eastward component. The northwest direction indicates that the northward component of the wind is equal to the eastward component.
Analyzing Upstream Movement
When the boat travels upstream, it moves against the river's current. The effective velocity in this case is:
- Upstream velocity: Vb - Vr = 3 m/s - 1 m/s = 2 m/s (westward)
Here, the flag points northeast, suggesting that the wind is again influencing the boat's direction. The northeast direction indicates that the northward component of the wind is equal to the eastward component, but in this case, the boat is moving westward while still being pushed northward by the wind.
Setting Up the Equations
Let’s denote the eastward component of the wind as VwE and the northward component as VwN. From the observations:
- For downstream (northwest direction): VwN = VwE
- For upstream (northeast direction): VwN = VwE
Since the boat's effective velocity downstream is 4 m/s and upstream is 2 m/s, we can set up the following equations based on the vectors:
- Downstream: 4 = VwE + 1
- Upstream: 2 = VwE - 1
Solving the Equations
From the downstream equation:
From the upstream equation:
Both equations confirm that the eastward component of the wind is 3 m/s. Since the northward component is equal to the eastward component, we can conclude that:
Final Analysis of Options
Now, let's evaluate the options provided:
- (A) Eastward component of velocity of wind is 1 - Incorrect
- (B) Northward component of velocity of wind is 1 - Incorrect
- (C) Eastward component of velocity of wind is 3 - Correct
- (D) Northward component of velocity of wind is 3 - Correct
Thus, the correct answers are (C) and (D), indicating that both the eastward and northward components of the wind's velocity are 3 m/s. This analysis shows how the boat's movement and the wind's influence can be understood through vector addition and directionality.
