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Chapter 6: Problem 17

Adult cheetahs, the fastest of the great cats, have a mass of about 70 kg and have been clocked to run at up to 72 mi/h (32 m/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) 35,840 Joules, (b) factor of 4.

Step by step solution

01

Understanding Kinetic Energy

Kinetic energy (KE) is the energy that an object possesses due to its motion. The formula to calculate the kinetic energy of an object is given by: \[ KE = \frac{1}{2} m v^2 \]where \( m \) is the mass of the object in kilograms (kg), and \( v \) is its velocity in meters per second (m/s).
02

Calculating Initial Kinetic Energy

First, let's substitute the mass (\( m = 70 \) kg) and velocity (\( v = 32 \) m/s) of the cheetah into the kinetic energy formula:\[ KE = \frac{1}{2} \times 70 \times (32)^2 \]Calculate \( v^2 \):\[ 32^2 = 1024 \]Now, substitute back into the formula:\[ KE = \frac{1}{2} \times 70 \times 1024 = 35 \times 1024 = 35840 \text{ Joules} \]
03

Understanding How Kinetic Energy Changes with Speed

The kinetic energy depends on the square of the velocity. So, if the velocity is doubled, the kinetic energy changes according to the square of the factor by which the velocity is increased.
04

Doubling the Velocity

To see how the kinetic energy changes when the velocity is doubled, substitute the new velocity, which is \( 2v = 2 \times 32 = 64 \) m/s, into the kinetic energy formula:\[ KE_{new} = \frac{1}{2} m (2v)^2 \]\[ KE_{new} = \frac{1}{2} \times 70 \times (64)^2 \]Calculate \( (64)^2 \):\[ 64^2 = 4096 \]So,\[ KE_{new} = \frac{1}{2} \times 70 \times 4096 = 35 \times 4096 = 143360 \text{ Joules} \]
05

Finding the Factor of Change in Kinetic Energy

To find the factor by which the kinetic energy increases, divide the new kinetic energy by the initial kinetic energy:\[ \text{Factor} = \frac{KE_{new}}{KE} = \frac{143360}{35840} \]Calculate the factor:\[ \text{Factor} = 4 \]

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Key Concepts

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

Velocity
Velocity describes how fast something is moving and specifies its direction. It is essential for calculating kinetic energy because it's one of the main variables in the kinetic energy formula. Velocity is measured in meters per second (m/s), indicating how many meters an object travels per second in a specified direction.
When discussing velocity, it is important to remember:
  • Velocity is a vector quantity, meaning it has both magnitude (speed) and direction.
  • In the kinetic energy equation, velocity is squared. This means that even small changes in velocity result in significant changes in kinetic energy.
  • For the cheetah running at its full speed, its velocity was 32 m/s.
Understanding these basics of velocity helps to grasp why doubling the velocity quadruples the kinetic energy, as kinetic energy depends on velocity raised to the power of two, making it highly sensitive to changes in speed.
Mass
Mass is a measure of the amount of matter in an object, measured in kilograms (kg). It is another key factor in the calculation of kinetic energy, as indicated by its presence in the kinetic energy formula: \[ KE = \frac{1}{2} m v^2 \]Here, mass is represented as \( m \).
The significance of mass in kinetic energy is highlighted by:
  • Mass is a scalar quantity, meaning it only has magnitude and no direction.
  • In the kinetic energy formula, mass is directly proportional to kinetic energy. Therefore, doubling the mass doubles the kinetic energy, assuming velocity stays the same.
  • For our cheetah example, the mass was given as 70 kg, showing us that heavier objects at the same speed have more kinetic energy.
Considering mass helps us comprehend how different animals or objects with varying weights have different kinetic energies even when moving at the same velocity.
Energy Transformation
Energy transformation is the process by which energy changes from one form to another. In the context of a cheetah running, chemical energy stored in its muscles is transformed into kinetic energy as it accelerates.
Energy transformations are crucial for several reasons:
  • Kinetic energy is the energy of motion, derived from the chemical energy in food converted by the body.
  • As speed increases, the transformation results in increased kinetic energy, highlighting the efficiency of energy conversion in animals.
  • If velocity doubles, as explored in the exercise, the kinetic energy increases fourfold, showcasing how energy transformations can exponentially increase with speed changes due to the squared relationship in the kinetic energy formula.
This concept helps in understanding not only how animals use and transform energy to move, but also illustrates basic principles of physics where energy is always conserved, just changed from one form to another in dynamic processes.

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Most popular questions from this chapter

Two blocks are connected by a very light string passing over a massless and frictionless pulley (\(\textbf{Fig. E6.7}\)). Traveling at constant speed, the 20.0-N block moves 75.0 cm to the right and the 12.0-N block moves 75.0 cm downward. How much work is done (a) on the 12.0-N block by (i) gravity and (ii) the tension in the string? (b) How much work is done on the 20.0-N block by (i) gravity, (ii) the tension in the string, (iii) friction, and (iv) the normal force? (c) Find the total work done on each block.

A tandem (two-person) bicycle team must overcome a force of 165 N to maintain a speed of 9.00 m/s. Find the power required per rider, assuming that each contributes equally. Express your answer in watts and in horsepower.

Use the work\(-\)energy theorem to solve each of these problems. You can use Newton's laws to check your answers. Neglect air resistance in all cases. (a) A branch falls from the top of a 95.0-m-tall redwood tree, starting from rest. How fast is it moving when it reaches the ground? (b) A volcano ejects a boulder directly upward 525 m into the air. How fast was the boulder moving just as it left the volcano?

A proton with mass 1.67 \(\times\) 10\(^{-27}\) kg is propelled at an initial speed of 3.00 \(\times\) 10\(^5\) m/s directly toward a uranium nucleus 5.00 m away. The proton is repelled by the uranium nucleus with a force of magnitude \(F = \alpha/x^2\), where \(x\) is the separation between the two objects and \(\alpha = 2.12 \times 10^{-26} \, \mathrm{N} \cdot \mathrm{m}^2\). Assume that the uranium nucleus remains at rest. (a) What is the speed of the proton when it is \(8.00 \times 10^{-10}\) m from the uranium nucleus? (b) As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the proton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? (c) What is the speed of the proton when it is again 5.00 m away from the uranium nucleus?

A mass \(m\) slides down a smooth inclined plane from an initial vertical height \(h\), making an angle \(\alpha\) with the horizontal. (a) The work done by a force is the sum of the work done by the components of the force. Consider the components of gravity parallel and perpendicular to the surface of the plane. Calculate the work done on the mass by each of the components, and use these results to show that the work done by gravity is exactly the same as if the mass had fallen straight down through the air from a height \(h\). (b) Use the work\(-\)energy theorem to prove that the speed of the mass at the bottom of the incline is the same as if the mass had been dropped from height \(h\), independent of the angle \(\alpha\) of the incline. Explain how this speed can be independent of the slope angle. (c) Use the results of part (b) to find the speed of a rock that slides down an icy frictionless hill, starting from rest 15.0 m above the bottom.
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