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How high can we jump? The maximum height a typical human can jump from a crouched start is about \(60 \mathrm{~cm} .\) By how much does the gravitational potential energy increase for a \(72 \mathrm{~kg}\) person in such a jump? Where does this energy come from?

Short Answer

Expert verified
The gravitational potential energy increases by 423.36 J, coming from the person's muscle energy.

Step by step solution

01

Understanding Gravitational Potential Energy

Gravitational potential energy (GPE) is given by the formula \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point. This formula helps us calculate the potential energy gained when an object is elevated to a certain height.
02

Identify Known Values

From the problem, we know the mass \( m \) of the person is \(72 \mathrm{~kg}\), the acceleration due to gravity \( g \) is approximately \(9.81 \mathrm{~m/s^2}\), and the maximum height \( h \) the person can jump is \(0.60 \mathrm{~m}\).
03

Apply the Formula

Substitute the known values into the gravitational potential energy formula: \( PE = 72 \mathrm{~kg} \times 9.81 \mathrm{~m/s^2} \times 0.60 \mathrm{~m} \).
04

Calculate the Potential Energy Increase

Perform the calculation: \[ PE = 72 \times 9.81 \times 0.60 = 423.36 \mathrm{~J} \]. Therefore, the gravitational potential energy increases by approximately \(423.36\) joules.
05

Source of Increased Energy

The energy required for this increase in potential energy comes from the conversion of chemical energy stored in the person's muscles. This energy is transformed during the act of jumping.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Education
Physics education involves teaching and learning about the laws and principles governing the physical world around us. This encompasses understanding concepts like gravitational potential energy, energy conservation, and mechanical energy. When learning physics, students engage with equations and problem-solving exercises that challenge them to apply these concepts.
Gravitational potential energy (GPE) is a crucial idea often explored in physics classes. It tells us how much energy is stored in an elevated object due to its position in a gravitational field. Using the formula \( PE = mgh \), where \( m \) is mass, \( g \) is gravity, and \( h \) is height, students learn to calculate the potential energy of an object at various heights.
Physics education is not only about solving formulas but also about understanding the underlying principles that govern how the universe functions. This exercise about jumping offers a practical glimpse into how humans interact with physical forces.
Energy Conversion
Energy conversion is the process of changing one form of energy into another. In our exercise, we see chemical energy within muscles being transformed into gravitational potential energy as a person jumps. This conversion is an example of how energy can shift forms to perform work.
In everyday life, we witness energy conversion all the time, such as electrical energy turning into light in a bulb or kinetic energy being converted to thermal energy as an object slows down. The body's ability to translate stored chemical energy into mechanical energy emphasizes the efficiency of biological processes.
  • During a jump, chemical energy stored from food intake powers muscular contractions.
  • These contractions generate a force that propels the body upward, overcoming gravitational pull.
  • As the person ascends to 0.60 m, the chemical energy is fully converted into gravitational potential energy, totaling around 423.36 J.
Understanding energy conversion helps us recognize the adaptability and interconnectedness of various energy forms in both natural and technological systems.
Mechanical Energy
Mechanical energy is the sum of kinetic energy (energy of motion) and potential energy (energy of position). It reflects the ability of an object to do work based on its motion and position. In the jumping exercise, the person initially uses kinetic energy to become airborne, then shifts to having only potential energy at the peak.
While ascending, the kinetic energy decreases as gravitational potential energy increases. When the height is maximum, kinetic energy momentarily becomes zero, and gravitational potential energy reaches its highest value. This interplay illustrates energy conservation, where the total mechanical energy (kinetic plus potential) remains constant (assuming no air resistance or other losses).
  • Before the jump, most energy is chemical, waiting to convert to kinetic as the muscles prepare.
  • As the person leaves the ground, energy conversion continues seamlessly between kinetic and potential energy.
  • At the jump's apex, all mechanical energy manifests as potential energy.
This exercise demonstrates the principles of mechanical energy, revealing how energy changes states seamlessly in a physical action like jumping.
Human Kinetics
Human kinetics studies the mechanics of body movements, focusing on how muscles and joints work to produce motion. The jumping exercise showcases a simple yet powerful display of human kinetics.
When a person jumps, muscles contract, stored chemical energy is converted, and movement occurs. Human kinetics helps us understand the biomechanics behind this movement, including muscle activity, force generation, and bodily coordination.
Explaining it relates to:
  • The energy stored in leg muscles is strategically released to optimize the height of the jump.
  • The coordinated effort involves hip, knee, and ankle joints working in harmony.
  • Knowing how muscles work can optimize physical performance and guide safer movements in sports and exercise contexts.
Understanding human kinetics and the energy transformations seen here can lead to a greater appreciation of human physical capabilities and improve techniques for athletic and everyday activities. This insight is critical in fields like sports science and rehabilitation.

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Most popular questions from this chapter

Fleas are agile, wingless insects that feed on the blood of their hosts. Although they are typically \(2-3 \mathrm{~mm}\) long with a mass of \(4.5 \times 10^{-4} \mathrm{~kg},\) they have an astonishing ability to jump when threatened. Their propulsion, which can briefly produce accelerations more than 100 times that of gravity, comes not from muscles but, in fact, from an elastomeric protein called resilin, which acts as a spring. Given that the typical launch velocity of a flea is about \(1 \mathrm{~m} / \mathrm{s},\) what total energy must be stored in the resilin just before the flea jumps?

A spring of negligible mass has force constant \(k=1600 \mathrm{~N} / \mathrm{m}\). (a) How far must the spring be compressed so that \(3.20 \mathrm{~J}\) of potential energy is stored in it? (b) You place the spring vertically with one end on the floor. You then drop a \(1.20 \mathrm{~kg}\) book onto it from a height of \(0.80 \mathrm{~m}\) above the top of the spring. Find the maximum distance the spring will be compressed.

On an essentially frictionless horizontal ice-skating rink, a skater moving at \(3.0 \mathrm{~m} / \mathrm{s}\) encounters a rough patch that reduces her speed by \(45 \%\) due to a friction force that is \(25 \%\) of her weight. Use the work-energy theorem to find the length of the rough patch.

A spring with spring constant \(k\) is anchored to the wall on one side of a hockey rink. A hockey puck is pressed against the spring and then released to slide across the ice. In the process the hockey puck gains a kinetic energy \(K .\) Derive an expression for the initial compression of the spring \(x\) in terms of \(k\) and \(K\).

A certain spring stores \(10.0 \mathrm{~J}\) of potential energy when it is stretched by \(2.00 \mathrm{~cm}\) from its equilibrium position. (a) How much potential energy would the spring store if it were stretched an additional \(2.00 \mathrm{~cm} ?\) (b) How much potential energy would it store if it were compressed by \(2.00 \mathrm{~cm}\) from its equilibrium position? (c) How far from the equilibrium position would you have to stretch the string to store \(20.0 \mathrm{~J}\) of potential energy? (d) What is the force constant of this spring?

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