A typical car battery stores about \(1 \mathrm{kW}\). h of energy. What's its power output if it drains completely in (a) 1 minute, (b) 1 hour, and (c) 1 day?

When we talk about energy storage in terms of car batteries, we're referring to the battery's capacity to retain and deliver electrical power as needed. Car batteries, usually lead-acid type, are designed to provide a quick surge of electrical energy necessary for starting a car's engine.

This storage is typically measured in kilowatt-hours (kWh), a unit of energy that represents the power of one kilowatt operating for one hour. To visualize, a battery that can deliver 1 kilowatt (1,000 watts) of power steadily for one hour is said to have a capacity of 1 kWh. Illustratively, this is akin to keeping a 100-watt bulb lit for 10 hours. Energy density, often expressed in watt-hours per kilogram (Wh/kg), is another critical aspect of energy storage, indicating how much energy a battery can store relative to its weight.

The power output of a battery tells us how rapidly it can supply energy. It's calculated by dividing the energy the battery stores by the time it takes to release that energy. Keeping in mind that power is measured in watts (W), the power calculation becomes critical when determining how usable a battery is for a particular application.

To get the power output, we use the formula: \[ P = \frac{E}{t} \]
where \(P\) stands for power in watts, \(E\) is the energy in watt-seconds, and \(t\) is the time in seconds. A battery with a higher power output can deliver the same amount of energy in less time compared to one with a lower power output. This concept is vital for tasks requiring a substantial burst of energy quickly, such as starting a car.

Unit conversion is an essential process in physics and engineering that allows for clear communication and calculations. For car batteries and power calculations, we often need to convert from kilowatt-hours, a unit of energy, to watts, a unit of power, and then factor in the time in seconds to find the power output.

For example, converting kilowatt-hours to watt-seconds involves a two-step process. First, knowing that 1 kilowatt-hour is equivalent to 1,000 watts operating for 3,600 seconds (1 hour), we can find the total energy in watt-seconds. In the exercise, 1 kWh converts to 3,600,000 watt-seconds. Understanding this form of unit conversion is fundamental when calculating a battery's power output as it allows us to accurately represent the rate at which a battery can deliver energy.