Question

Suppose that your car averages \(30 \mathrm{mi} / \mathrm{gal}\) of gasoline. \((a)\) How far could you travel on \(1 \mathrm{~kW} \cdot \mathrm{h}\) of energy consumed? \((b)\) If you are driving at \(55 \mathrm{mi} / \mathrm{h}\), at what rate are you expending energy? The heat of combustion of gasoline is \(140 \mathrm{MJ} / \mathrm{gal}\).

Short answer

a) You can travel approximately 0.771 mi on 1 kWh of energy. b) You're expending energy at a rate of 71.3 kW when driving at 55 mi/h.

Step-by-step solution

Conversion of energy units and calculation of traveled distance

First, convert the energy from kWh to MJ: \(1 kWh = 3.6 MJ\). So if you have 1 kWh then you have \(1 kWh * 3.6 MJ/kWh = 3.6 MJ\). Next, use the car's fuel efficiency and the heat of combustion of gasoline to figure out how far you can travel on this energy. From the heat of combustion, you know that 1 gal of gasoline provides 140 MJ. So, 3.6 MJ of energy would be the equivalent of \(3.6 MJ * (1 gal / 140 MJ) = 0.0257 gal\). Then, use the fuel efficiency of the car to see how far you can travel on 0.0257 gal: \(0.0257 gal * (30 mi / 1 gal) = 0.771 mi\)

Calculation of energy expenditure rate

Next, you want to know how fast you're using energy when driving 55 mi/h. To figure this out, you need to calculate how much gasoline is used per hour: \(55 mi/h * (1 gal / 30 mi) = 1.833 gal/h\). Then, convert this gas usage rate to an energy usage rate: \(1.833 gal/h * (140 MJ/1 gal) = 256.67 MJ/h\)

Convert the rate of energy consumption to an appropriate unit

256.67 MJ/h is hard to interpret. It's more common to talk about power usage in kW. So, convert it! First, convert MJ to kJ: \(256.67 MJ/h * (1000 kJ / 1 MJ) = 256,670 kJ/h\). Then, convert kJ/h to kW: \(1 kW = 1 kJ/s\), and \(1 h = 3600 s\), so \(256,670 kJ/h * (1 kW / 3600 kJ) = 71.3 kW\)

Key concepts

Fuel Efficiency Calculation

Fuel efficiency is a critical factor for drivers and environmentalists alike, with impacts on both the wallet and the planet. It represents the distance a vehicle can travel on a specific amount of fuel and is usually stated in miles per gallon (mpg) in the United States or liters per 100 kilometers (L/100km) in other parts of the world.

To calculate how far you can travel on a given amount of energy, it's essential to understand the relationship between fuel efficiency, the heat of combustion of the fuel, and the energy content. For instance, in our example, a car that averages 30 mpg with gasoline, one can convert the energy provided by electricity (in this case, 1 kWh) to the equivalent amount of gasoline (in this instance, found to be 0.0257 gallons), and then multiply by the car's fuel efficiency to find the distance covered (0.771 miles).

This fuel efficiency calculation helps to compare the cost and environmental impact of different fuels and vehicles, highlighting which are more economical and produce fewer emissions per mile.

Heat of Combustion

The heat of combustion is a measure of the energy released when a given amount of fuel is burned. In physics and chemistry, it's defined as the amount of heat energy produced per unit mass or volume of the fuel. Typically, heat of combustion is expressed in units of energy such as megajoules per gallon (MJ/gal) or calories per gram.

In our exercise, gasoline has a heat of combustion of 140 MJ/gal. Knowing this value is pivotal in determining how much energy is available from the fuel and thus how far a vehicle can travel. When combined with the fuel efficiency of a car, you can calculate the energy required to travel a certain distance and, by extension, the equivalent distance that can be traveled using a non-fossil-fuel-based energy source (e.g., electricity).

This measure is not only useful for calculating travel distances but is also essential for engineers to design engines and for environmentalists to understand and mitigate the impact of consuming fossil fuels.

Power Usage in kW

Understanding Power in Kilowatts

Power is the rate at which energy is used, and in the context of vehicles, it is the rate at which fuel energy is converted to kinetic energy to move the car. Power is often measured in kilowatts (kW), especially when we discuss electricity usage or the power rating of electric motors.

One kilowatt equates to 1,000 joules of energy being used per second. To find out how much power is being used while driving, such as in our problem, you need to first determine the rate of energy consumption (in MJ/h or kJ/h) and then convert it to the more user-friendly unit of kilowatts. In part b of the exercise, we established that driving at 55 mph expends energy at a rate of 256.67 MJ/h. Converting that to kilowatts gave us a power usage rate of 71.3 kW, painting a clear picture of the vehicle's power requirements at that speed.

Knowing the power usage in kW not only helps in understanding vehicle performance but also in making more informed decisions about energy consumption, fuel economy, and even the sizing of home solar power systems.