Question

A consumer is comparing two new cars. Car A exerts 250 horsepower, while Car B exerts 300 horsepower. The consumer is most concerned about the peak velocity that the car can reach. If nonconservative forces can be ignored, which of the following statements is true? (Note: 1 horsepower \(=745.7 \mathrm{W}\) ) (A) Car A and Car B both have unlimited velocities. (B) Car A will reach its peak velocity more quickly than Car B. (C) Car A will dissipate less energy to the surroundings than Car B. (D) Car A will have a lower peak velocity than Car B.

Short answer

D: Car A will have a lower peak velocity than Car B.

Step-by-step solution

- Understand Power and Velocity Relationship

Recognize that power is related to the ability to do work over time, and for a car, it relates to the ability to reach a higher velocity. Higher power means the car can exert more energy per unit of time, allowing it to achieve a higher peak velocity under similar conditions.

- Convert Horsepower to Watts

Convert the horsepower of each car to watts to properly compare their power outputs. Car A has 250 hp, which is equivalent to $$250 \times 745.7 \text{ W} = 186425 \text{ W}.$$ Car B has 300 hp, which is equivalent to $$300 \times 745.7 \text{ W} = 223710 \text{ W}.$$ Thus, Car B has a higher power output.

- Analyze Peak Velocity

Since Car B has a higher power output than Car A, it will be able to achieve a higher peak velocity. As nonconservative forces are ignored, the power directly translates to how quickly each car can accelerate and the maximum speed they can reach.

- Evaluate Statements

Assess the given statements based on the above analysis:(A) Both cars do not have unlimited velocities since power limits their maximum speeds.(B) Car A will not reach its peak velocity more quickly since it has less power.(C) Energy dissipation to the surroundings is not relevant as nonconservative forces are ignored.(D) Car A will have a lower peak velocity than Car B as calculated.

- Conclusion

Based on the power-to-velocity relationship: (D) Car A will have a lower peak velocity than Car B.

Key concepts

Power and Velocity Relationship

When discussing the physics of vehicles, understanding the relationship between power and velocity is essential. Power is defined as the rate at which work is done or energy is transferred. For cars, higher power translates to a greater ability to reach and sustain high velocities. Therefore, a vehicle with higher horsepower can exert more energy per unit of time, resulting in a higher peak velocity. This relationship shows why cars with more powerful engines generally perform better in terms of speed, especially when other factors like drag and friction are negligible.

Horsepower Conversion

Horsepower is a common unit used to measure the power of engines. To compare different engines accurately, it's often necessary to convert horsepower to watts. One horsepower is equal to 745.7 watts. For example, if Car A has 250 horsepower, its power in watts would be calculated as follows:

250 hp \(\times 745.7 \text{ W/hp} = 186425 \text{ W}\).

Similarly, for Car B with 300 horsepower:

300 hp \(\times 745.7 \text{ W/hp} = 223710 \text{ W}\).

This conversion helps in directly comparing the power outputs, showing that Car B has a significantly higher power output than Car A.

Peak Velocity Dynamics

The peak velocity of a car is the maximum speed it can achieve under given conditions. With higher power output, a car can overcome inertia faster, meaning it can reach and sustain higher speeds more easily. This is particularly true when nonconservative forces, such as friction and air resistance, are ignored. In our example exercise, because Car B has a greater power output than Car A, it can reach a higher peak velocity. This direct implication of higher power allowing for superior speed performance can be observed consistently across various applications in automotive physics.

Energy Dissipation

Energy dissipation involves the loss of energy, typically due to nonconservative forces such as friction, air resistance, and heat. In many real-world scenarios, energy dissipation significantly affects the performance of a vehicle. However, in this exercise, we assume that nonconservative forces are negligible. This means that all the energy produced by the car's engine is used to accelerate the car, with no loss to surroundings. Under this idealized condition, the car's power completely determines its peak velocity. Therefore, energy dissipation is not a factor in comparing the velocities of Car A and Car B in this specific context.