Problem 49 - University Physics with Modern Physics

Question

How many joules of energy does a \(100-\) watt light bulb use per hour? How fast would a 70 -kg person have to run to have that amount of kinetic energy?

Step-by-Step Solution

Answer

The 100-watt bulb uses 360,000 joules per hour. A 70-kg person must run at about 101.42 m/s to have the same kinetic energy.

Step 1: Understanding Energy Use

A watt measures power, which is energy per unit of time. Specifically, a watt equals one joule per second. Thus, a 100-watt bulb uses 100 joules of energy every second.

Step 2: Calculate Energy Used per Hour

To find out how many joules the bulb uses in an hour, we need to calculate the energy use over that period. Since there are 3600 seconds in an hour, multiply the power by the number of seconds: \(100 \text{ watts} \times 3600 \text{ seconds} = 360000 \text{ joules}\).

Step 3: Understanding Kinetic Energy Formula

The formula for kinetic energy (KE) is \( KE = \frac{1}{2} m v^2 \), where \(m\) is mass in kilograms, and \(v\) is velocity in meters per second. This represents the energy an object has due to its motion.

Step 4: Set Up the Equation

We know the kinetic energy must equal the energy the bulb uses in one hour, so set \( KE = 360000 \text{ joules} \). Plug in the person's mass \( m = 70 \text{ kg} \): \( 360000 = \frac{1}{2} \times 70 \times v^2 \).

Step 5: Solve for Velocity

Rearrange the kinetic energy formula to solve for velocity \(v\):\[360000 = 35v^2 \v^2 = \frac{360000}{35} \v^2 \approx 10285.71 \v \approx \sqrt{10285.71} \v \approx 101.42 \] Thus, the velocity is approximately 101.42 meters per second.

Key Concepts

kinetic energy

Kinetic energy is the energy an object possesses due to its motion. When an object moves, it has the capacity to do work, which is directly related to its mass and velocity. The formula for kinetic energy is given by: \[ KE = \frac{1}{2} mv^2 \] where:

• \( KE \) is the kinetic energy
• \( m \) is the mass of the object in kilograms
• \( v \) is the velocity of the object in meters per second

This equation shows that kinetic energy increases with the square of the velocity; thus, small increases in velocity can result in large increases in kinetic energy. This principle is crucial in understanding the physical impact of moving objects. For a 70 kg person, to match the energy used by a 100-watt bulb in an hour, they would need to run at a specific velocity. By rearranging the formula, we can solve for the required speed, which provides practical insight into the kinetic energy concept.

power and energy

Power and energy are closely related, yet distinct concepts. Power refers to the rate at which energy is used or transferred, measured in watts (W).

• One watt equals one joule per second. Hence, power tells us how much energy a device uses per second.

In contrast, energy is the total amount used over time and is measured in joules (J). These concepts are fundamental when analyzing devices like a 100-watt light bulb, which consumes 100 joules every second. Over an hour, this amounts to a significant total energy use, calculated by multiplying the power by time (in seconds). Understanding how power relates to energy helps us manage electricity usage and improve efficiency. Daily, it gives rise to practical applications such as measuring the total electricity consumed in households. By evaluating wattage and usage time, one can estimate their energy consumption and potentially reduce costs through more efficient energy practices.

joules to watt-hour conversion

The conversion between joules and watt-hours is an essential skill for understanding energy usage in practical terms. Since many devices state energy consumption in kilowatt-hours (kWh), comprehending how to convert from joules helps you grasp this measurement.To convert joules to watt-hours, understand that:

• 1 watt-hour equals 3600 joules, as one watt is one joule/second and there are 3600 seconds in an hour.

Therefore, to determine how many watt-hours corresponds to a certain number of joules, divide the total joules by 3600.For instance, a 100-watt bulb that uses 360,000 joules in an hour is equivalent to \(\frac{360000}{3600} = 100\) watt-hours. This method helps you compare energy consumption across various appliances easily, especially when evaluating cost or efficiency based on household power bills. Making this conversion part of everyday considerations can aid in actionable improvements toward more sustainable energy consumption habits.