/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 111 A \(9.40 \mathrm{~kg}\) projecti... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 111

A \(9.40 \mathrm{~kg}\) projectile is fired vertically upward. Air drag decreases the mechanical energy of the projectile Earth system by \(68.0 \mathrm{~kJ}\) during the projectile's ascent. How much higher would the projectile have gone were air drag negligible?

Short Answer

Expert verified
The projectile would have gone approximately 737.18 meters higher without air drag.

Step by step solution

01

Understanding the problem

We need to find out how much higher the projectile would have gone if there was no air drag affecting its ascent. The drag reduces the mechanical energy of the system by 68.0 kJ.
02

Conversion of energy units

Convert the mechanical energy from kilojoules to joules to use it in further calculations: \[ 68.0 \text{ kJ} = 68,000 \text{ J} \]
03

Calculate the energy used to oppose gravity

The lost mechanical energy equals the gravitational potential energy (GPE) that could have been gained without air drag, which is given by:\[ \Delta E = mgh \]where \( m = 9.40 \text{ kg} \) is the mass, \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity, and \( h \) is the additional height gained. Rearranging gives:\[ h = \frac{\Delta E}{mg} \]
04

Substitute and solve for h

Substituting the known values into the equation:\[ h = \frac{68,000 \text{ J}}{9.40 \text{ kg} \times 9.81 \text{ m/s}^2} \]Calculate\[ h \approx \frac{68,000}{92.214} \approx 737.18 \text{ m} \]
05

Interpret the result

The projectile would have ascended approximately 737.18 meters higher if air drag were negligible.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mechanical Energy
Mechanical energy is the sum of two types of energy: kinetic and potential. It plays a crucial role in projectile motion. When a projectile is fired vertically, the initial mechanical energy is mainly kinetic due to its motion. As the projectile ascends, some of this kinetic energy is converted into gravitational potential energy.
  • Kinetic Energy: This is the energy that an object possesses due to its motion, given by the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.
  • Potential Energy: This involves energy stored due to an object's position, such as height in a gravitational field.
Understanding the interplay of these energies helps us analyze energy conversion and conservation in systems, especially when forces like air drag are involved.
Gravitational Potential Energy
When a projectile moves upward, its gravitational potential energy (GPE) increases. This energy depends on the height relative to a reference point, usually the ground beneath the projectile. The formula for GPE is \( GPE = mgh \), where \( m \) stands for mass, \( g \) for the gravitational acceleration (about 9.81 m/s虏 near the Earth's surface), and \( h \) is the height.
  • Energy at Peak: At the highest point, the projectile's GPE is at a maximum, while kinetic energy is minimal since velocity is zero at the peak.
  • Energy Conversion: Any initial kinetic energy converted entirely into GPE determines the peak height.
Calculating the potential energy loss due to air drag involves understanding how higher potential energy could reach without this energy loss.
Air Drag
Air drag, or air resistance, is a force acting opposite to the relative motion of any object moving through air. In projectile motion, air drag dissipates energy, reducing both speed and the achievable height of the projectile.
  • Energy Dissipation: It converts mechanical energy into thermal energy, dissipating it into the surroundings.
  • Effects on Motion: Less remaining energy means a reduced ascent height, translating into less gravitational potential energy gained.
Understanding air drag is critical when calculating maximum reach, helping to interpret scenarios where actual performance deviates from ideal predictions.
Energy Conversion
Energy conversion in projectile motion involves transitioning between kinetic and potential energies. Without air drag, these conversions are seamless, and total mechanical energy remains constant throughout the flight.
  • Conservation of Energy: Ideally, in the absence of air resistance, mechanical energy at the launch should equal that at every energy state during the ascent and descent.
  • Real-World Application: Air drag disrupts this conservation by converting some mechanical energy into other forms, reducing the energy that can be converted to height. This explains why the absence of air drag results in a greater ascent, as shown in the solved problem.
Hence, understanding energy conversion clarifies how various factors like air drag affect projectiles, leading to more accurate predictions and analyses.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In a circus act, a \(60 \mathrm{~kg}\) clown is shot from a cannon with an initial velocity of \(16 \mathrm{~m} / \mathrm{s}\) at some unknown angle above the horizontal. A short time later the clown lands in a net that is \(3.9 \mathrm{~m}\) vertically above the clown's initial position. Disregard air drag. What is the kinetic energy of the clown as he lands in the net?

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?

A \(50 \mathrm{~g}\) ball is thrown from a window with an initial velocity of \(8.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(30^{\circ}\) above the horizontal. Using energy methods, determine (a) the kinetic energy of the ball at the top of its flight and (b) its speed when it is \(3.0 \mathrm{~m}\) below the window. Does the answer to (b) depend on either (c) the mass of the ball or (d) the initial angle?

A \(15 \mathrm{~kg}\) block is accelerated at \(2.0 \mathrm{~m} / \mathrm{s}^{2}\) along a horizontal frictionless surface, with the speed increasing from \(10 \mathrm{~m} / \mathrm{s}\) to \(30 \mathrm{~m} / \mathrm{s} .\) What are \((\mathrm{a})\) the change in the block's mechanical energy and (b) the average rate at which energy is transferred to the block? What is the instantaneous rate of that transfer when the block's speed is (c) \(10 \mathrm{~m} / \mathrm{s}\) and (d) \(30 \mathrm{~m} / \mathrm{s} ?\)

A spring with spring constant \(k=620 \mathrm{~N} / \mathrm{m}\) is placed in a vertical orientation with its lower end supported by a horizontal surface. The upper end is depressed \(25 \mathrm{~cm},\) and a block with a weight of \(50 \mathrm{~N}\) is placed (unattached) on the depressed spring. The system is then released from rest. Assume that the gravitational potential energy \(U_{g}\) of the block is zero at the release point \((y=0)\) and calculate the kinetic energy \(K\) of the block for \(y\) equal to \((\mathrm{a}) 0,(\mathrm{~b}) 0.050 \mathrm{~m},(\mathrm{c}) 0.10 \mathrm{~m},(\mathrm{~d}) 0.15 \mathrm{~m},\) and \((\mathrm{e}) 0.20 \mathrm{~m} .\) Also (f) how far above its point of release does the block rise?
See all solutions

Recommended explanations on Physics Textbooks

Space Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Electricity and Magnetism

Read Explanation

Scientific Method Physics

Read Explanation

Wave Optics

Read Explanation

Physical Quantities And Units

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.