Introduction:

When a moving car encounters air resistance, it requires power to overcome this resistance and maintain a constant speed. The magnitude of air resistance depends on the speed of the car. In this scenario, the air resistance is directly proportional to the square of the speed of the car.

Formula for air resistance:

The air resistance on the car can be represented by the equation:

Air resistance = k * v²

where v is the speed of the car and k is the constant of proportionality.

Power required:

The power required to overcome the air resistance can be calculated using the formula:

Power = Force * Velocity

Since the force is given by the air resistance, we can substitute it into the formula:

Power = Air resistance * Velocity

Comparison at 40 km/h and 80 km/h:

To compare the power required at 40 km/h and 80 km/h, we need to calculate the air resistance at each speed and then calculate the power based on that.

Calculating air resistance:

For 40 km/h:
- Convert the speed to m/s: 40 km/h = 40 * 1000 / 3600 = 11.11 m/s
- Calculate the air resistance: Air resistance = k * (11.11)²

For 80 km/h:
- Convert the speed to m/s: 80 km/h = 80 * 1000 / 3600 = 22.22 m/s
- Calculate the air resistance: Air resistance = k * (22.22)²

Calculating power:

Now, we can calculate the power required at each speed using the formula:

Power = Air resistance * Velocity

For 40 km/h:
- Power = Air resistance at 40 km/h * 11.11 m/s

For 80 km/h:
- Power = Air resistance at 80 km/h * 22.22 m/s

Ratio of power:

To find the ratio of the power required at 40 km/h to that required at 80 km/h, we divide the power at 40 km/h by the power at 80 km/h:

Ratio = (Power at 40 km/h) / (Power at 80 km/h)

Explanation:

When we calculate the ratio of the power required at 40 km/h to that required at 80 km/h, we can observe that the ratio is not 1:2.

Since the air resistance is proportional to the square of the speed, the power required to overcome the air resistance increases more than proportionally with the speed. This means that doubling the speed will require more than double the power.

In this case, the ratio of the power required at 40 km/h to that required at 80 km/h will be greater than 1:2 due to the squared relationship between air resistance and speed.