Question

A 220-lb man jumps out a window into a fire net \(36 \mathrm{ft}\) below. The net stretches \(4.4 \mathrm{ft}\) before bringing him to rest and tossing him back into the air. What is the potential energy of the stretched net?

Short answer

The potential energy of the stretched net is approximately 1313.78 Joules.

Step-by-step solution

Convert pounds to kilograms

Firstly, it is necessary to convert the man's weight from pounds to kilograms, because the standard unit in physics for weight is the kilogram. 1 pound is approximately equal to 0.453592 kg. So, the weight of the man in kilograms is \(220 \times 0.453592 = 99.79 \, \mathrm{kg}\) (rounded to two decimal points).

Convert feet to meters

Similar to the weight, the lengths have to be converted from feet to meters. 1 foot is approximately equal to 0.3048 meter. Therefore, the distance the man falls is \(36 \times 0.3048 = 10.97 \, \mathrm{m}\) (rounded to two decimal points). Further, the distance the net stretches is \(4.4 \times 0.3048 = 1.34 \, \mathrm{m}\) (rounded to two decimal points).

Calculate the potential energy

Potential energy, usually symbolized by \(U\), is calculated using the formula \(U = m \times g \times h\), where \(m\) represents the mass (in kg), \(g\) is the gravitational acceleration (which is approximately \(9.81 \, \mathrm{m/s^2}\) on the surface of the Earth), and \(h\) indicates height (in m). In this scenario, the height is the distance the net stretches when the man falls into it. Plugging in the values gives \(U = 99.79 \times 9.81 \times 1.34 = 1313.78 \, \mathrm{J}\) (rounded to two decimal points). Thus, the potential energy of the stretched net is approximately 1313.78 Joules.

Key concepts

Energy Conversion

Energy conversion is a key concept in physics as it describes how energy changes from one form to another. In the scenario provided, the man’s gravitational potential energy as he is about to fall is converted into kinetic energy as he descends. When he lands on the net and it stretches, the kinetic energy is then converted to elastic potential energy in the net.

Understanding energy conversion is essential because energy does not get created or destroyed, according to the principle of conservation of energy. Instead, energy continuously transforms, allowing for various physical phenomena to occur.

Consider:

• When potential energy decreases during the fall, kinetic energy increases equivalently.
• The net absorbs energy, slowly converting kinetic energy into potential energy.

This energy transformation is combated by forces like friction and air resistance in real-life scenarios, which can convert some energy into heat as well.

Gravitational Potential Energy

Gravitational potential energy (GPE) is a type of potential energy associated with the position of an object within a gravitational field. Before he jumps, the man possesses gravitational potential energy due to his elevated position above the ground.

GPE can be calculated using the formula: \[ U = m \times g \times h \]
where:

• \( m \) = mass of the object (in kilograms),
• \( g \) = acceleration due to gravity (usually \( 9.81 \, \mathrm{m/s^2} \) on Earth),
• \( h \) = height from which the object falls or is elevated (in meters).

Using this formula, we can find the energy due to the man’s height above the net as potential energy due to gravity aligns with how far he might fall.

Unit Conversion

Unit conversion is essential in physics to ensure that calculations are performed correctly and accurately using the standard units of measurement. In the provided problem, the man’s weight in pounds is converted to kilograms, and distances in feet are converted to meters. This allows for using the International System of Units (SI units) to calculate potential energy.

The conversion steps are as follows:

• Mass: Multiply pounds (\(\mathrm{lbs}\)) by 0.453592 to obtain kilograms (\(\mathrm{kg}\)).
• Distance: Multiply feet (\(\mathrm{ft}\)) by 0.3048 to obtain meters (\(\mathrm{m}\)).

Ensuring the correct units is crucial in physics to avoid erroneous calculations and to maintain consistency and accuracy across physical equations. Knowing how to convert between different units helps solve various mathematical and scientific problems effectively.