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Chapter 8: Problem 47

A \(75 \mathrm{~g}\) Frisbee is thrown from a point \(1.1 \mathrm{~m}\) above the ground with a speed of \(12 \mathrm{~m} / \mathrm{s}\). When it has reached a height of \(2.1 \mathrm{~m}\), its speed is \(10.5 \mathrm{~m} / \mathrm{s}\). What was the reduction in \(E_{\mathrm{mec}}\) of the Frisbee-Earth system because of air drag?

Short Answer

Expert verified
The reduction in mechanical energy is approximately \(0.4725 \, \text{J}\).

Step by step solution

01

Write the formula for mechanical energy

The mechanical energy (E_{mec}) of the Frisbee is the sum of its kinetic energy (KE) and potential energy (PE). We can writeE_{mec} = KE + PE = \frac{1}{2} m v^2 + mgh, where \(m\) is the mass of the Frisbee, \(v\) is its velocity, and \(h\) is its height above the ground.
02

Calculate the initial mechanical energy

Initially, the Frisbee is at a height of \(1.1 \, \text{m}\) with a speed of \(12 \, \text{m/s}\). Calculate the initial KE and PE: \[\text{KE}_{\text{initial}} = \frac{1}{2} \cdot 0.075 \, \text{kg} \cdot (12 \, \text{m/s})^2\] and \[\text{PE}_{\text{initial}} = 0.075 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 1.1 \, \text{m}\]. Find \(\text{E}_{\text{mec, initial}}\) by summing these energies.
03

Calculate the final mechanical energy

At a height of \(2.1 \, \text{m}\), the Frisbee's speed is \(10.5 \, \text{m/s}\). Calculate the final KE and PE: \[\text{KE}_{\text{final}} = \frac{1}{2} \cdot 0.075 \, \text{kg} \cdot (10.5 \, \text{m/s})^2\] and \[\text{PE}_{\text{final}} = 0.075 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 2.1 \, \text{m}\]. Find \(\text{E}_{\text{mec, final}}\) by summing these energies.
04

Calculate the reduction in mechanical energy

Determine the change in mechanical energy due to air drag by finding the difference between the initial and final mechanical energies: \[\Delta E_{\text{mec}} = E_{\text{mec, final}} - E_{\text{mec, initial}}\]. The reduction in mechanical energy is the absolute value of \(\Delta E_{\text{mec}}\).

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Key Concepts

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

Understanding Kinetic Energy
Kinetic energy is the energy that an object possesses due to its motion. Any moving object has kinetic energy, and its amount depends on two factors: the object's mass and its velocity. The formula for kinetic energy (KE) is given by the equation: \[ KE = \frac{1}{2} mv^2 \] where \(m\) is the mass of the object and \(v\) is its velocity.
  • If the velocity is increased, the kinetic energy increases exponentially, because velocity is squared in the formula.
  • Even a small increase in speed can lead to a significant rise in kinetic energy.
  • Conversely, if the object's speed decreases, its kinetic energy will also decrease.
In the case of the Frisbee, its kinetic energy decreases from its initial toss to when it reaches a height of 2.1 meters. This decrease in speed as the Frisbee moves higher decreases its kinetic energy, which helps in understanding the concept of energy transformation in motion.
Potential Energy in Action
Potential energy is the stored energy of an object due to its position or configuration. For objects positioned above the ground, this is most commonly gravitational potential energy. It can be calculated using the formula: \[ PE = mgh \] Here, \(m\) is the mass, \(g\) is the acceleration due to gravity (approximately 9.8 m/sdataon Earth), and \(h\) is the height above the reference point (for example, the ground).
  • As an object is elevated higher, its potential energy increases.
  • Likewise, lowering the object would decrease its potential energy.
  • This type of energy is known as potential because it has the potential to be converted into kinetic energy or other forms of energy.
When the Frisbee climbs from 1.1 mto 2.1 m, its potential energy rises because it is further against the force of gravity, showing another fascinating aspect of energy conversion in mechanical systems.
Role of Air Resistance
Air resistance, also known as drag, is a force that acts against the motion of an object as it moves through air. It plays a crucial role in energy dynamics by affecting the object's speed and energy transformation.
  • As a result of air resistance, objects face opposition when moving through the air, which can cause them to slow down.
  • This deceleration leads to a reduction in kinetic energy, which, as our original exercise suggests, results in reduced mechanical energy.
  • Understanding air resistance is critical for calculations involving moving objects as it can change the balance of energy by dissipating mechanical energy in the form of heat and other forms.
In the Frisbee scenario, air resistance is responsible for some of the mechanical energy loss. As the Frisbee moves through the air, it experiences a drag force, which results in the speed reduction and, consequently, a drop in its overall mechanical energy.

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Most popular questions from this chapter

From the edge of a cliff, a \(0.55 \mathrm{~kg}\) projectile is launched witl an initial kinetic energy of \(1550 \mathrm{~J}\). The projectile's maximum up ward displacement from the launch point is \(+140 \mathrm{~m}\). What are the (a) horizontal and (b) vertical components of its launch velocity? (c) At the instant the vertical component of its velocity is \(65 \mathrm{~m} / \mathrm{s}\), what is its vertical displacement from the launch point?

A locomotive with a power capability of \(1.5 \mathrm{MW}\) can accelerate a train from a speed of \(10 \mathrm{~m} / \mathrm{s}\) to \(25 \mathrm{~m} / \mathrm{s}\) in \(6.0 \mathrm{~min}\). (a) Calculate the mass of the train. Find (b) the speed of the train and (c) the force accelerating the train as functions of time (in seconds) during the \(6.0\) min interval. (d) Find the distance moved by the train during the interval.

What is the spring constant of a spring that stores \(25 \mathrm{~J}\) of elastic potential energy when compressed by \(7.5 \mathrm{~cm} ?\)

The only force acting on a particle is conservative force \(\vec{F}\). If the particle is at point \(A\), the potential energy of the system associated with \(\vec{F}\) and the particle is \(40 \mathrm{~J}\). If the particle moves from point \(A\) to point \(B\), the work done on the particle by \(\vec{F}\) is \(+25 \mathrm{~J}\). What is the potential energy of the system with the particle at \(B ?\)

A child whose weight is \(267 \mathrm{~N}\) slides down a \(6.1 \mathrm{~m}\) playground slide that makes an angle of \(20^{\circ}\) with the horizontal. The coefficient of kinetic friction between slide and child is \(0.10 .\) (a) How much energy is transferred to thermal energy? (b) If she starts at the top with a speed of \(0.457 \mathrm{~m} / \mathrm{s}\), what is her speed at the bottom?
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