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.
The problem states that the force required to row a boat at constant velocity 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.

### Step 2: Relate power to force and speed.
Power is defined as the product of force and velocity:
\[ P = F \cdot v \]
Substituting the relationship from Step 1, we have:
\[ P \propto v^2 \cdot v = v^3 \]
This means power is proportional to the cube of the speed.

### 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 \( 2v \). According to the relationship we derived:
\[ \frac{P_1}{P_2} = \frac{v^3}{(2v)^3} \]

### Step 4: Simplify the ratio.
Calculating \( (2v)^3 \):
\[ (2v)^3 = 8v^3 \]
Thus, the ratio becomes:
\[ \frac{P_1}{P_2} = \frac{v^3}{8v^3} = \frac{1}{8} \]

### Step 5: Express \( P_2 \) in terms of \( P_1 \).
From the ratio, we can express \( P_2 \):
\[ P_2 = 8P_1 \]

### Step 6: Substitute the known power value.
We know that at speed \( v \), the power \( P_1 \) is 4 kW:
\[ P_2 = 8 \times 4 \text{ kW} = 32 \text{ kW} \]

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

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