The force required to row a boat at constant velocity is proportional to square of its speed. If a speed of v km/h requires 4 kW, how much power does a speed of 2v km/h require?

Options

A. 8 kW
B. 16 kW
C. 24 kW
D. 32 kW

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between force, speed, and power
We know that the force required to row the boat is proportional to the square of its speed. This can be expressed mathematically as:
\[ F \propto v^2 \]
where \( F \) is the force and \( v \) is the speed of the boat.

### Step 2: Relate power to force and speed
Power (\( P \)) is defined as the product of force and velocity:
\[ P = F \cdot v \]
Since we have established that \( F \propto v^2 \), we can substitute this into the power equation:
\[ P \propto v^2 \cdot v = v^3 \]
This means that power is proportional to the cube of the speed:
\[ P \propto v^3 \]

### Step 3: Set up the ratio of powers at different speeds
Let \( P_1 \) be the power required at speed \( v \) and \( P_2 \) be the power required at speed \( 2v \). From the proportionality established, we can write:
\[ \frac{P_2}{P_1} = \left(\frac{2v}{v}\right)^3 = 2^3 = 8 \]

### Step 4: Substitute the known power value
We know from the problem that at speed \( v \), the power \( P_1 = 4 \) kW. Thus, we can substitute this value into our ratio:
\[ P_2 = 8 \cdot P_1 = 8 \cdot 4 \text{ kW} \]

### Step 5: Calculate the power at speed \( 2v \)
Now we calculate:
\[ P_2 = 8 \cdot 4 = 32 \text{ kW} \]

### Conclusion
The power required to row the boat at a speed of \( 2v \) km/h is **32 kW**.

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