The force required to tow a car at constant velocity is directly proportional to velocity. If it requires 160 W to two a car with a velocity of `4ms^(-1)`, the power it required to move the car with a velocity of `9ms^(-1)` is

Options

A. 900W
B. 180W
C. 810 W
D. 640 W

Answer

To solve the problem, we need to determine the power required to tow a car at a different velocity, given that the force required is directly proportional to the velocity. We'll follow these steps:

### Step 1: Understand the relationship between power, force, and velocity
The power \( P \) required to tow an object is given by the formula:
\[ P = F \cdot v \]
where \( F \) is the force and \( v \) is the velocity.

### Step 2: Establish the proportional relationship
Since the force required to tow the car is directly proportional to the velocity, we can express this as:
\[ F \propto v \]
This implies that:
\[ F = k \cdot v \]
for some constant \( k \).

### Step 3: Relate power to velocity
Substituting \( F \) in the power formula, we get:
\[ P = (k \cdot v) \cdot v = k \cdot v^2 \]
This shows that power is directly proportional to the square of the velocity:
\[ P \propto v^2 \]

### Step 4: Set up the ratio of powers
For two different velocities \( v_1 \) and \( v_2 \), we can write:
\[ \frac{P_1}{P_2} = \frac{v_1^2}{v_2^2} \]
where \( P_1 \) and \( P_2 \) are the powers at velocities \( v_1 \) and \( v_2 \) respectively.

### Step 5: Insert known values
From the problem, we know:
- \( P_1 = 160 \, \text{W} \)
- \( v_1 = 4 \, \text{m/s} \)
- \( v_2 = 9 \, \text{m/s} \)

Plugging these values into the ratio gives:
\[ \frac{160}{P_2} = \frac{4^2}{9^2} \]

### Step 6: Calculate the squares
Calculating the squares:
\[ 4^2 = 16 \quad \text{and} \quad 9^2 = 81 \]
Thus, we have:
\[ \frac{160}{P_2} = \frac{16}{81} \]

### Step 7: Cross-multiply to solve for \( P_2 \)
Cross-multiplying gives:
\[ 160 \cdot 81 = 16 \cdot P_2 \]
\[ 12960 = 16 \cdot P_2 \]

### Step 8: Solve for \( P_2 \)
Now, divide both sides by 16:
\[ P_2 = \frac{12960}{16} = 810 \, \text{W} \]

### Final Answer
The power required to move the car with a velocity of \( 9 \, \text{m/s} \) is:
\[ \boxed{810 \, \text{W}} \]