/*! 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 14 \(\cdot\) \(\cdot\) Animal energ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 14

\(\cdot\) \(\cdot\) Animal energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 \(\mathrm{kg}\) and have been clocked at up to 72 mph \((32 \mathrm{m} / \mathrm{s})\) . (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

Short Answer

Expert verified
(a) 35840 J; (b) Factor of 4.

Step by step solution

01

Understanding Kinetic Energy Formula

Kinetic energy is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. We'll use this formula to calculate the kinetic energy of the cheetah.
02

Calculate Kinetic Energy at Original Speed

First, we need to plug in the values for the cheetah's mass \( m = 70 \; \mathrm{kg} \) and speed \( v = 32 \; \mathrm{m/s} \) into the kinetic energy formula. \[KE = \frac{1}{2} \times 70 \times (32)^2 \]Calculate this to find the kinetic energy.
03

Compute the Expression

Let's compute the expression: \[ KE = \frac{1}{2} \times 70 \times 1024 = 35 \times 1024 \]\[ KE = 35840 \; \mathrm{J} \]So, the kinetic energy of the cheetah is \( 35840 \; \mathrm{J} \).
04

Calculate Kinetic Energy at Double Speed

To find the kinetic energy when the speed is doubled, first determine the new speed: New speed = \( 2 \times 32 = 64 \; \mathrm{m/s} \).Plug into the formula:\[KE_{new} = \frac{1}{2} \times 70 \times (64)^2 \]Calculate \( KE_{new} \).
05

Compute Double Speed Expression

Compute the expression for the doubled speed:\[ KE_{new} = \frac{1}{2} \times 70 \times 4096 = 35 \times 4096 \]\[ KE_{new} = 143360 \; \mathrm{J} \]The kinetic energy at double speed is \( 143360 \; \mathrm{J} \).
06

Determine the Factor of Change

Divide the new kinetic energy by the original kinetic energy to determine the factor of change:\[ \text{Factor} = \frac{143360}{35840} = 4 \]The kinetic energy increases by a factor of 4 when the speed is doubled.

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.

Kinetic Energy Formula
Understanding the Kinetic Energy Formula is key to grasping how moving objects store energy. The formula for kinetic energy (KE) is given by:
  • \(KE = \frac{1}{2} m v^2 \)
Here, \( m \) represents the mass of the object, and \( v \) is its velocity. This formula reveals that kinetic energy depends on two factors: mass and speed. A slight increase in speed results in a greater increase in kinetic energy since velocity is squared in the formula.
Given that mass remains constant, doubling the speed increases the kinetic energy by a factor of four, illustrating the speed's significant impact on kinetic energy.
Physics Problem Solving
Solving physics problems involves systematically applying formulas to find unknown values. Start by understanding what is being asked, identifying the information given, and deciding on the appropriate formula to use. For the cheetah problem, we need to determine the kinetic energy at various speeds.
This process requires calculating kinetic energy at the original speed first. Then, you expand this process to solve for kinetic energy when the speed is doubled. Each step builds on the results of the previous one, which is a common strategy in physics problem solving.
Energy Transformation
Energy transformation refers to the change of energy from one form to another. In moving objects like a cheetah, chemical energy from food is transformed into kinetic energy. This transformation allows the animal to move swiftly.
When a cheetah accelerates, more chemical energy is turned into kinetic energy, illustrating the direct correlation between energy transformation and speed. Understanding this aids in grasping broader principles of energy conservation and applicability in the real world.
Speed and Velocity
Speed and velocity are crucial components in understanding how kinetic energy functions. Speed is the rate at which an object covers distance, while velocity includes both speed and direction. In the kinetic energy formula, it is the speed that is considered, but knowing the direction (velocity) is important for full comprehension in physics contexts.
For instance, in the given exercise, the cheetah reaches a speed of 32 m/s. If this speed is vectorized to include direction, it becomes velocity. However, since kinetic energy depends on the magnitude of speed, we focus purely on the scalar value for calculations.
Mathematical Calculations in Physics
Mathematical Calculations in Physics are central to understanding and predicting the behavior of physical systems. Using the kinetic energy formula involves straightforward multiplication and squaring of numbers. In our example with the cheetah, calculating kinetic energy requires multiplying mass by the square of velocity, then taking half of that product.
These calculations provide quantitative insight into energy transformations and help predict outcomes when variables change, such as doubling the speed and observing the effect on kinetic energy. Through consistent practice, students can develop proficiency in these calculations, which are foundational to deeper exploration in physics.

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

\(\bullet\) \(\bullet\) You and three friends stand at the corners of a square whose sides are 8.0 \(\mathrm{m}\) long in the mid- dle of the gym floor, as shown in the accompanying figure. You take your physics book and You take your physics book and push it from one person to the other. The book has a mass of \(1.5 \mathrm{kg},\) and the coefficient of kinetic friction between the book and the floor is \(\mu_{k}=0.25\) . (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connect- ing these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim who then slides it back to you. What is the total work done by fric- tion during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain.

\(\bullet\) \(\bullet\) At the site of a wind farm in North Dakota, the average wind speed is \(9.3 \mathrm{m} / \mathrm{s},\) and the average density of air is 1.2 \(\mathrm{kg} / \mathrm{m}^{3} .\) (a) Calculate how much kinetic energy the wind contains, per cubic meter, at this location. (b) No wind turbine can capture all of the energy contained in the wind, the main reason being that capturing all the energy would require stop- ping the wind completely, meaning that air would stop flowing through the turbine. Suppose a particular turbine has blades with a radius of 41 \(\mathrm{m}\) and is able to capture 35\(\%\) of the avail- able wind energy. What would be the power output of this tur- bine, under average wind conditions?

\(\bullet\) \(\bullet\) Food calories. The food calorie, equal to \(4186 \mathrm{J},\) is a measure of how much energy is released when food is metabo- lized by the body. A certain brand of fruit-and-cereal bar con- tains 140 food calories per bar. (a) If a 65 \(\mathrm{kg}\) hiker eats one of these bars, how high a mountain must he climb to "work off" the calories, assuming that all the food energy goes only into increasing gravitational potential energy? (b) If, as is typical, only 20\(\%\) of the food calories go into mechanical energy, what would be the answer to part (a)? (Note: In this and all other problems, we are assuming that 100\(\%\) of the food calories that are eaten are absorbed and used by the body. This actually not true. A person's "metabolic efficiency" is the percentage of calories eaten that are actually used; the rest are eliminated by the body. Metabolic efficiency varies considerably from person to person.)

\(\cdot\) \(\cdot\) A racing dog is initially running at \(10.0 \mathrm{m} / \mathrm{s},\) but is slow- ing down. (a) How fast is the dog moving when its kinetic energy has been reduced by half? (b) By what fraction has its kinetic energy been reduced when its speed has been reduced by half?

\(\bullet\) \(\bullet\) \(\bullet\) Mass extinctions. One of the greatest mass extinc- tions occurred about 65 million years ago, when, along with many other life-forms, the dinosaurs went extinct. Most geologists and paleontologists agree that this event was caused when a large asteroid hit the earth. Scientists esti- mate that this asteroid was about 10 \(\mathrm{km}\) in diameter and that it would have been traveling at least as fast as 11 \(\mathrm{km} / \mathrm{s}\) . The density of asteroid material is about \(3.5 \mathrm{g} / \mathrm{cm}^{3},\) on the aver- age. (a) What would be the approximate mass of the aster- oid, assuming it to be spherical? (b) How much kinetic energy would the asteroid have delivered to the earth? (c) In perspective, consider the following: the total amount of energy used in one year by the human race is roughly 500 exajoules (see Appendix E). If this rate of energy use remained constant, how many years would it take the human species to use an amount of energy equal to the amount delivered by this asteroid?
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Physics of Motion

Read Explanation

Nuclear Physics

Read Explanation

Rotational Dynamics

Read Explanation

Classical Mechanics

Read Explanation

Medical Physics

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.