Question

How much power, in horsepower, must be developed by the engine of a \(1600-\mathrm{kg}\) car moving at \(26 \mathrm{~m} / \mathrm{s}(=94 \mathrm{~km} / \mathrm{h})\) on a level road if the forces of resistance total \(720 \mathrm{~N}\) ?

Short answer

The engine must develop approximately 25.1 horsepower

Step-by-step solution

Identify given variables

The mass of the car \( m \) is 1600 kg, it is moving with a speed \( v \) of 26 m/s, and the total forces of resistance \( F \) is 720 N.

Compute the Power in Watts

The formula for power is \( P = F \times v \) . Substituting the given values into the formula, we get: \( P = 720 \, N \times 26 \, m/s = 18720 \, W \).

Convert Power from Watts to Horsepower

To convert the power from watts to horsepower, we use the conversion factor of 1 horsepower = 746 watts. So, \( P_{hp} = P_W / 746 \). Substituting the earlier calculated power into the formula, we get: \( P_{hp} = 18720 \, W / 746 \approx 25.1 \, hp \).

Key concepts

Power in Watts

Understanding the concept of power in watts is critical in physics, especially in problems relating to energy conversion and mechanical work. Power refers to the rate at which work is done or the rate at which energy is transferred or converted. It is a measure of how quickly a force can move an object over a certain distance. In the International System of Units (SI), power is measured in watts (W).

One watt is defined as the rate of work when an object is moved at a speed of one meter per second against a force of one newton. In mathematical terms, power (P) can be calculated using the formula:
\[ P = F \times v \]
where \( F \) is the force in newtons, and \( v \) is the velocity in meters per second. For example, if a car's engine is exerting a force of 720 N to maintain a speed of 26 m/s, it is developing a power of:\[ P = 720 \, N \times 26 \, m/s = 18720 \, W \]
which is a sizable amount of power used to overcome the forces of resistance and maintain the car's speed.

Horsepower

Horsepower (hp) is another unit of power commonly used in the automotive industry and refers to the power output of engines, including car engines. It originates from the historical comparison of engine output to the amount of work a horse can do. Although not an SI unit, horsepower is still widely used, especially in the United States.

To convert from the SI unit of power, which is watts, to horsepower, one can use the conversion factor:\[ 1 \, hp = 746 \, W \]
Using this conversion factor, the power in watts can be converted to horsepower as follows:\[ P_{hp} = \frac{P_{W}}{746} \]
In the example of the car's engine developing a power of 18720 W, the power in horsepower would be:\[ P_{hp} = \frac{18720 \, W}{746} \approx 25.1 \, hp \]
This value represents the engine's capacity to perform work similar to that amount of horses, thus giving a layperson a relatable measure of the vehicle's capabilities.

Forces of Resistance

Forces of resistance play a significant role in mechanical systems and vehicular dynamics. They are forces that oppose motion, such as friction, air resistance (drag), and rolling resistance in vehicles. These forces require power from the engine to overcome them to maintain a constant velocity.

When calculating work or power, it's essential to consider the resistance forces, as they directly influence the amount of power necessary for an object to continue moving at a constant speed. Generally, the total resistance force (F) can include multiple factors and is the sum of various forces resisting the movement. In our case of a moving car, the forces of resistance can include:

• Friction between the car tires and the road
• Air resistance acting against the car's body
• Internal mechanical resistance in the car's components

For the exercise, the forces of resistance are given as 720 N. Thus, the engine needs to produce enough power not only to move the car but also to overcome these forces. This is why understanding resistance is crucial for solving problems related to power and energy efficiency in vehicles.