Question

An engine expends 40.0 hp in moving a car along a level track at a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). How large is the total force acting on the car in the opposite direction of the motion of the car?

Short answer

Answer: The total force acting on the car in the opposite direction of its motion is approximately 1,988.54 N.

Step-by-step solution

Convert horsepower to watts

First, we need to convert the power from horsepower (hp) to watts (W). Use the conversion factor 1 hp = 745.7 W. Multiply 40.0 hp by 745.7 W/hp to get the power in watts. \(40.0 \, \text{hp} \times 745.7 \, \frac{\text{W}}{\text{hp}} = 29,828 \, \text{W}\)

Calculate force

Now that the power is in watts, we'll use the formula for power to find the force acting in the opposite direction of the motion: Power = Force × Speed We know the power (29,828 W) and the speed (15.0 m/s). Therefore, we can rewrite the formula to solve for force: Force = Power / Speed Plug in the values: Force = \(\frac{29,828 \, \text{W}}{15.0 \, \text{m/s}} = 1,988.54 \, \text{N}\)

Present the final result

The total force acting on the car in the opposite direction of its motion is approximately 1,988.54 N.

Key concepts

Engine Power Calculation

Engine power is a crucial measure of how much work an engine can perform over time. In physics, power is the rate at which work is done, and for engines, this is often measured in horsepower (hp).
Horsepower is a traditional unit of power that was adopted in the automotive industry due to historical reasons. However, in scientific calculations, we usually convert it to watts (W), which is the standard unit of power in the International System of Units (SI).
For example, in our exercise, an engine provides 40.0 hp of power. Using the conversion factor where 1 hp equals 745.7 W, we convert horsepower to watts to facilitate easier calculations. This conversion
resulted in an engine power of 29,828 W, which gives us a better understanding in the context of physics calculations.

• Remember: 1 hp = 745.7 W
• Convert power units for easier calculations

Force and Motion

Force and motion are interconnected in physics, as described by Newton's Second Law of Motion. This law states that force equals mass times acceleration (F = ma). In our scenario, force is also associated with the way the engine moves the car along the track.
As the engine provides power, it generates force to counteract resistive forces such as friction and air resistance.
This force acts in the opposite direction to maintain constant motion, ensuring that the car continues to move at a steady speed. By calculating the force, we gain insights into the engine's ability to sustain motion against resistance or drag.

• Force is needed to maintain motion
• Engine force counters resistance

Unit Conversion

Unit conversion is a fundamental skill in physics, essential for ensuring all measurements are in consistent units.
In our problem, we encounter units like horsepower, which need to be converted into watts to facilitate easy calculation of force and power using standard formulas.
Additionally, speed is already given in the standard unit of meters per second (m/s), making calculations straightforward once other units have been adjusted accordingly.
When performing unit conversions, it’s important to be precise and remember conversion factors for accuracy in calculations. This ensures that the result we compute, such as the total force, is reliable and clearly understood.

• Keep track of units
• Use proper conversion factors
• Accuracy in conversion is key

Power Formula

The power formula is fundamental to solving problems related to engine performance and motion. Power is defined as the rate at which work is done, and it can be calculated using the formula:

\[ \text{Power} = \text{Force} \times \text{Speed} \]
By rearranging this formula, we can solve for force, which was our goal. Using the given power in watts and the speed of the car in meters per second (m/s), we divided the power by speed to find the force acting in the opposite direction of motion.

\[ \text{Force} = \frac{\text{Power}}{\text{Speed}} \]
This straightforward calculation helps us understand the relationship between power, force, and speed, and how they influence each other in physical systems.

• Understand the relation: Power, Force & Speed
• Formula: \( \text{Power} = \text{Force} \times \text{Speed} \)
• Rearrange for desired solution

Newton's Laws of Motion

Newton's Laws of Motion are fundamental principles that describe how objects move. In the exercise, the focus is mainly on the second law: Force equals mass times acceleration (F = ma).
However, a broader understanding incorporates all three laws. For constant velocity, as described in our problem, the first law is also applicable. It states that an object in motion stays in motion with a constant velocity unless acted upon by a net external force.
The resistive forces here are countered by the engine's power output, resulting in a balance that maintains steady motion. The understanding of these laws aids in comprehending how an engine’s force counteracts resistance and keeps the car moving smoothly.

• First Law: Object's motion remains constant unless there's an external force
• Second Law: Focus on calculating the force needed to maintain motion
• Third Law: Action-reaction forces balance