To solve this problem, we need to determine the swimmer's speed relative to the water ($v_s$) that will allow them to swim directly from point A to point B, compensating for the current.

Given Data:

• $d_1 = 200 \, \text{m}$ (distance from A to C)
• $d_2 = 150 \, \text{m}$ (distance from C to B)
• $v_r = 5 \, \text{km/h}$ (speed of the river current)
• $\theta = 45^\circ$ (angle at which swimmer swims relative to the line from A to C)

Step 1: Convert speeds to consistent units.

The current speed is given in km/h. We need to convert this to m/s: $v_r = 5 \, \text{km/h} = \frac{5 \times 1000}{3600} \, \text{m/s} = 1.3889 \, \text{m/s}$

Step 2: Resolve swimmer's velocity components.

The swimmer's velocity relative to the water $v_s$ has components:

• A horizontal component $v_s \cos(\theta)$
• A vertical component $v_s \sin(\theta)$

We want to ensure that the swimmer swims directly from A to B. The current will push the swimmer downstream, so the swimmer's horizontal component of velocity must cancel the effect of the river current.

Thus, the horizontal component of the swimmer's velocity relative to the water must equal the speed of the current: $v_s \cos(45^\circ) = v_r$

Since $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, we have: $v_s \cdot \frac{\sqrt{2}}{2} = 1.3889 \, \text{m/s}$ $v_s = \frac{1.3889 \times 2}{\sqrt{2}} \, \text{m/s}$ $v_s \approx \frac{2.7778}{1.4142} \, \text{m/s}$ $v_s \approx 1.962 \, \text{m/s}$

Step 3: Convert the swimmer's speed to km/h.

To convert from m/s to km/h, multiply by 3.6: $v_s = 1.962 \, \text{m/s} \times 3.6 \, \text{km/h per m/s}$ $v_s \approx 7.07 \, \text{km/h}$

Final Answer:

The swimmer's speed relative to the water should be approximately 7.07 km/h.

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Would you like more details or have any questions?

Here are 5 related questions:

1. How would the swimmer's speed change if the angle $\theta$ were different?
2. What would happen if the current speed were doubled?
3. How do you determine the swimmer's velocity if they swim at an angle other than 45°?
4. How could we find the time it would take the swimmer to cross the river?
5. How would the distances $d_1$ and $d_2$ change if the swimmer were not compensating for the current?

Tip: When dealing with currents in river problems, always break the swimmer’s velocity into components, especially when dealing with angles. This makes it easier to handle the directionality of the current.