Question

If an \(80 \mathrm{~kg}\) person is capable of delivering external mechanical energy at a rate of \(200 \mathrm{~W}\) sustained over several minutes, \({ }^{34}\) how high would they be able to climb in two minutes?

Short answer

Answer: The person can climb approximately 30.58 meters high in two minutes.

Step-by-step solution

Write down known variables

In the given exercise, we know the following variables: mass (m) = 80 kg power (P) = 200 W time (t) = 2 minutes = 120 seconds (conversion to seconds) The height at which the person can climb (h) needs to be determined.

Understand the concept of work and power

We must recall the relationship between work and power to solve this problem. Work (W) is equal to the product of force and distance, whereas power (P) is the rate at which work is done. The formulas are: Work (W) = Force (F) x Distance (d) Power (P) = Work (W) / Time (t) In this case, the distance is the height the person can climb, and the force is due to gravity acting on the person.

Relate force to work

The force acting on the person due to gravity is given by F = m * g where 'm' is the mass of the person and 'g' is the acceleration due to gravity (approximately 9.81 m/s^2). The work done while climbing is given by W = F * h, where 'h' is the height climbed. Combining these equations, we get: W = m * g * h

Calculate the work done, using power and time

We already know that Power (P) = Work (W) / Time (t), so we can rearrange the equation to find the work done: Work (W) = P * t

Substitute work from Step 3 into the equation from Step 4

By substituting the work from Step 3 into the equation from Step 4, we get: m * g * h = P * t Now, we will solve for the height (h): h = (P * t) / (m * g)

Plug in known values and solve for height

Finally, substitute the known values into the equation for height: h = (200 W * 120 s) / (80 kg * 9.81 m/s^2) h ≈ 30.58 meters The person can climb approximately 30.58 meters high in two minutes.

Key concepts

Work and Energy

Work and energy are two fundamental concepts in physics that are closely interconnected. Work is done when a force causes an object to move. It is calculated as the product of force and distance in the direction of the force. Represented mathematically as \(W = F \times d\), where \(W\) is the work done, \(F\) is the force applied, and \(d\) is the distance over which the force is applied. In the context of our problem, the work done by the person is due to the gravitational force over a certain height.

Energy, on the other hand, is the ability to do work. When a person is climbing, they are converting their metabolic energy into gravitational potential energy. The more they work, the higher their potential energy will be when they reach a certain height. Gravitational potential energy is given by the formula \(U = m \times g \times h\), where \(U\) is the potential energy, \(m\) is the mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.

Relating Work to Energy

As the person climbs, the work they do against gravity is directly converted into potential energy. We can see that the work done to climb a certain height (which involves overcoming the gravitational force) is equal to the gravitational potential energy gained. Using the relation \(W = U\), the exercise shows how understanding these concepts allows us to solve for the height the person can achieve by knowing the amount of energy (work) they can provide.

Mechanical Power

Mechanical power is the rate at which work is done or energy is transferred. In essence, it measures how quickly energy is being used or work is being performed. The unit of power is the Watt (W), with one Watt equating to one Joule per second. Mathematically, it is expressed as \(P = \frac{W}{t}\), where \(P\) is power, \(W\) is work, and \(t\) is time.

In our exercise, the person's power output is given as 200 W, which means they can continuously do 200 Joules of work every second. Understanding the power output allows us to determine the amount of work the person can accomplish over a given period of time. By rearranging the formula to \(W = P \times t\), we determine how much work the person can do in two minutes, which ultimately helps us find out how high they can climb given their power output.

Application of Power in the Solution

The step by step solution of the exercise effectively uses the concept of mechanical power to relate the rate of energy output by the person to the total work done over the time interval. We find that over a span of 120 seconds, the person is able to do a certain amount of work, which when equated to the work required to climb against gravity, reveals the maximum height that can be reached.

Gravitational Force

The gravitational force is an attractive force that acts between all masses. In the context of the provided physics problem, the gravitational force is what opposes the person's climb. The force of gravity near the Earth's surface is essentially constant and is represented by the acceleration due to gravity \(g\), with a standard value of approximately \(9.81 \text{ m/s}^2\). The force exerted by gravity on an object is equal to its mass multiplied by the acceleration due to gravity \(F = m \times g\).

The work done to lift the person against gravity depends on this force. The greater the mass or the greater the height, the more work is needed to perform the climb. This is precisely why heavier objects require more energy to lift and why it takes more work to climb to higher elevations.

Calculating Height Using Gravitational Force

In our example, multiplying the force of gravity by the height obtained from the power and work equations allows us to solve for the distance the person can climb. This outcome highlights the importance of the gravitational force in work and energy problems where vertical motion against gravity is involved.