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 direction opposite to the motion of the car?

Short answer

Answer: The total force acting on the car in the direction opposite to its motion is approximately 1,989.33 N.

Step-by-step solution

Convert horsepower to watts

First, we need to convert the engine power from horsepower to watts. 1 hp is equal to 746 watts. So, 40 hp can be converted to watts by multiplying it with the conversion factor: $$40.0 \ \text{hp} \times \frac{746 \ \text{watts}}{1 \ \text{hp}} = 29,840 \ \text{watts}$$

Use the power formula

The formula for power is: $$\text{Power} = \text{Force} \times \text{Velocity}$$ We need to find the force acting on the car, so we can rearrange the formula to solve for Force: $$\text{Force} = \frac{\text{Power}}{\text{Velocity}}$$

Substitute the given values

Now, we can substitute the given values into the formula: $$\text{Force} = \frac{29,840 \ \text{watts}}{15.0 \ \text{m/s}}$$

Calculate the force

By dividing the power by the velocity, we can find the force acting on the car: $$\text{Force} = \frac{29,840}{15} \approx 1,989.33 \ \text{N}$$ So, the total force acting on the car in the direction opposite to the motion is approximately 1,989.33 N.

Key concepts

Power to Force Calculation

Understanding the relationship between power and force is an essential aspect of physics problem-solving, particularly when analyzing mechanical systems like engines and vehicles. Power, in a physical context, refers to the rate at which work is done or energy is transferred. It is represented in watts (W) in the International System of Units (SI). When we relate power to force, we're essentially determining how much force is exerted over a certain distance within a specific timeframe.

To extract force from power, you need to know the velocity at which the object is moving. The formula tied to this concept is: \[\[\begin{align*}\text{Power} &= \text{Force} \times \text{Velocity}\end{align*}\]\]. By rearranging the formula to solve for Force, it becomes: \[\[\begin{align*}\text{Force} &= \frac{\text{Power}}{\text{Velocity}}\end{align*}\]\]. This rearrangement isolates the force as the subject of the equation, enabling us to determine how much force is responsible for the power output presented.

Horsepower to Watts Conversion

In many real-world applications, especially those related to automotive and mechanical engineering, power is commonly measured in horsepower (hp). One horsepower is the amount of energy it takes to lift 550 pounds one foot in one second and is equal to approximately 746 watts. The conversion from horsepower to watts is straightforward:

\[\[\begin{align*}\text{Watts} &= \text{Horsepower} \times 746\end{align*}\]\]. For instance, converting 40 horsepower to watts involves multiplying 40 by 746, yielding 29,840 watts. This conversion is crucial when you're working with power specifications not provided in the standard SI unit of watts. It's essential to operate within a common unit to apply the physical formulas correctly for problem-solving.

Force and Motion Analysis

The analysis of force and motion is a cornerstone of classical mechanics. Newton's second law of motion is foundational here, describing how the acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to its mass. Fundamentally, force is what causes an object to accelerate, slow down, or continue moving at a constant speed despite resistance, such as friction.

Within the context of the exercise provided, the total force acting on the car in the direction opposite to the motion—often referred to as drag force—can be calculated by examining the power exerted by the engine and the constant velocity of the car. Understanding how force interacts with motion allows for the determination of various other physical properties, such as the amount of work done over a distance, the energy consumed or produced, and the overall efficiency of mechanical systems.