Problem 51 - 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

A 100-watt bulb uses 360,000 joules per hour. A 70 kg person must run at 101.42 m/s to match that energy in kinetic form.

Step 1: Convert Watts to Joules

To find how many joules the bulb uses per hour, we start with its power rating: 100 watts. Since 1 watt = 1 joule/second, a 100-watt bulb uses \(100 \text{ joules/second}\). To find the energy consumption in one hour, multiply by the number of seconds in an hour (3600):\[100 \text{ watts} \times 3600 \text{ seconds} = 360,000 \text{ joules}.\]

Step 2: Kinetic Energy Formula

The kinetic energy \(E_k\) of an object is given by the formula \(E_k = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is velocity. We need to find the velocity that gives a 70 kg person 360,000 joules of energy.

Step 3: Solve for Velocity

Rearrange the kinetic energy formula to solve for velocity \(v\): \[v = \sqrt{\frac{2E_k}{m}}.\] Substitute \(E_k = 360,000\) joules and \(m = 70\) kg: \[v = \sqrt{\frac{2 \times 360,000}{70}}.\] Calculate: \[v \approx \sqrt{10,285.71} \approx 101.42 \text{ m/s}.\]

Step 4: Conclusion

A 70 kg person needs to run at approximately 101.42 meters per second to have 360,000 joules of kinetic energy, the same as the energy used by a 100-watt light bulb in one hour.

Key Concepts

Energy Conversion

Energy conversion is the process of transforming energy from one form to another. In our exercise, we focus on converting electrical power, measured in watts, into energy, measured in joules. This is a common conversion since electrical appliances, like our 100-watt light bulb, have power ratings that help us understand their energy usage. Understanding energy conversion allows us to compare different forms of energy and see how they relate in practical applications.

Watts to Joules

Converting watts to joules can be straightforward once you know the basics. Watts measure power, which is the rate of energy use or generation. Joules, on the other hand, measure energy itself. One watt is equivalent to one joule per second.

To convert watts to joules for any duration, you multiply the watts by the number of seconds during which the energy flow is happening. So, in our example:

• The light bulb is 100 watts, meaning it uses 100 joules every second.
• There are 3600 seconds in an hour.
• Therefore, in one hour, the bulb uses 100 watts × 3600 seconds, equating to 360,000 joules.

This practical conversion process is key for understanding energy consumption in everyday devices.

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy in an object related to its motion and position. In this exercise, we are mainly concerned with kinetic energy.

Kinetic energy is the energy possessed due to motion and can be calculated using the formula: \[ E_k = \frac{1}{2}mv^2 \]

• \( m \) is mass
• \( v \) is velocity.

In our scenario with the person running, we focus entirely on kinetic energy to achieve the equivalent of the light bulb's energy. The more massive or faster an object, the greater its kinetic energy.

Understanding mechanical energy helps us appreciate how energy is harnessed and exerted, whether in machinery or living organisms.

Velocity Calculation

Calculation of velocity is essential when determining how fast an object must travel to possess a certain amount of kinetic energy. To solve for velocity in the context of kinetic energy, we rearrange the kinetic energy formula:\[ v = \sqrt{\frac{2E_k}{m}} \]In our exercise, \( E_k \) is 360,000 joules, and \( m \) is 70 kg.Substitute these values into the equation to find:\[ v = \sqrt{\frac{2 \times 360,000}{70}} \]This calculation gives us a velocity of approximately 101.42 meters per second.

This process shows how changes in kinetic energy reflect different speeds needed based on mass. It highlights the relationship between velocity and energy, illustrating that even small increases in speed can significantly impact energy in motion.