Question

If a 70 kg person (weight: \(700 \mathrm{~N}\) ) is capable of putting out energy at a rate of \(500 \mathrm{~W}\) in short bursts, how long will it take the person to race up a flight of stairs \(4 \mathrm{~m}\) high, considering only the vertical energy \(^{42}\) required?

Short answer

Answer: It will take the person 5.6 seconds to go up the flight of stairs.

Step-by-step solution

Calculate the gravitational potential energy (GPE) of the person

First, we need to calculate the GPE, which can be found by multiplying the person's weight by the height of the staircase. GPE = Weight × Height = \(700 \mathrm{~N} \times 4 \mathrm{~m} = 2800 \mathrm{~J}\)

Convert the power to Joules per second

Next, we need to convert the power from watts to Joules per second since watts are just an alternative way to express Joules per second. Power = \(500 \mathrm{~W} = 500 \mathrm{~J/s}\)

Calculate the time using work-energy principle

Now that we have the required energy (GPE) and the rate at which the person can provide the energy (Power), we can find how long it takes the person to make that work. By using the work-energy principle, we can derive the following formula: Time = GPE / Power Time = \(\frac{2800 \mathrm{~J}}{500\mathrm{~J/s}} = 5.6\mathrm{~s}\) It will take the person 5.6 seconds to race up the flight of stairs considering only the vertical energy required.

Key concepts

Gravitational Potential Energy

Gravitational Potential Energy, often abbreviated as GPE, is an essential concept in physics that describes the energy possessed by an object due to its position relative to Earth, or another massive body. It is the energy that is stored when an object is raised to a height against the force of gravity.
This energy can be calculated using the formula: \[ \text{GPE} = \text{Weight} \times \text{Height} \] For a person or object, the weight is the force due to gravity acting on them. For instance, in our problem, a 70 kg person on Earth has a weight of 700 N (newtons).
When this person moves up a 4 meter height, they gain gravitational potential energy of 2800 J (joules), calculated by multiplying their weight (700 N) by the height (4 m). Understanding GPE helps in calculating how much energy you need to "lift" an object to a certain height, which is important in various practical applications such as designing elevators or analyzing motion in mechanics.

Work-Energy Principle

The Work-Energy Principle is a powerful tool in physics that relates the work done on an object to the change in its energy. This principle is based on the idea that work done by forces on an object results in a change in the object’s energy.
For instance, if work is done to lift a person or move them upward against gravity, they gain potential energy, as seen when calculating GPE. The principle can be expressed with the equation: \[ \text{Work} = \text{Change in Energy} \]In our staircase exercise, the work done to climb is equal to the gravitational potential energy acquired by the person.
Therefore, Work = GPE (2800 J).
This helps us understand how forces and energy changes are intricately connected. By using this principle, we can relate how the exertion of forces (like the energy of climbing) impacts energy transformation, particularly in systems where mechanical energy is involved.

Power and Energy

Power and energy are two closely related concepts that often come up in physics and engineering. While energy is the capacity to do work or produce change, power is the rate at which energy is transferred or converted.
Power is expressed in watts (W), where one watt is equivalent to one joule per second. In the problem, the person is capable of exerting power at 500 W, meaning they can produce 500 joules of energy every second.
To determine the time it takes to climb the stairs, we use the formula: \[ \text{Time} = \frac{\text{Energy}}{\text{Power}} \]Applying this to our scenario, the person uses 2800 joules of energy to climb the stairs with a power output of 500 watts, resulting in a time of 5.6 seconds.
This exercise showcases how power dictates the pace of energy usage, especially useful in situations like athletic performance, where bursts of rapid energy are important. Understanding power enables us to calculate how effectively energy is utilized and the efficiency of different systems and tasks.