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Chapter 8: Problem 36

Compare the kinetic energy \(\left(=\frac{1}{2} m v^{2}\right)\) of a 1 -gram piece of ice (about half the mass of a dime) entering Earth's atmosphere at a speed of \(50 \mathrm{km} / \mathrm{s}\) to that of a 2 -metric-ton SUV (mass \(=2 \times\) \(10^{3} \mathrm{kg}\) speeding down the highway at \(90 \mathrm{km} / \mathrm{h}\).

Short Answer

Expert verified
The kinetic energy of the ice piece is 1,250,000 J, while that of the SUV is 625,000 J. The ice piece has more kinetic energy.

Step by step solution

01

- Convert Units for Ice Piece

Start by converting the mass of the ice piece into kilograms. Since 1 gram is equal to 0.001 kilograms, we have: \( m_{\text{ice}} = 0.001 \text{ kg} \)
02

- Calculate Kinetic Energy of Ice Piece

Using the formula for kinetic energy, substitute the mass and speed of the ice piece into the equation: \( KE_{\text{ice}} = \frac{1}{2} m_{\text{ice}} v_{\text{ice}}^{2} \)Given that the speed \(v_{\text{ice}} = 50 \text{ km/s} = 50,000 \text{ m/s} \): \( KE_{\text{ice}} = \frac{1}{2} \times 0.001 \text{ kg} \times (50,000 \text{ m/s})^{2} \) \( KE_{\text{ice}} = \frac{1}{2} \times 0.001 \text{ kg} \times 2,500,000,000 \text{ m}^{2}/\text{s}^{2} \) \( KE_{\text{ice}} = 1,250,000 \text{ J} \)
03

- Convert Speed for SUV

Convert the speed of the SUV from km/h to m/s. Since 1 km/h is equal to \( \frac{1}{3.6} \) m/s, we have: \( v_{\text{SUV}} = 90 \text{ km/h} = \frac{90}{3.6} \text{ m/s} = 25 \text{ m/s} \)
04

- Calculate Kinetic Energy of SUV

Using the formula for kinetic energy, substitute the mass and speed of the SUV into the equation: \( KE_{\text{SUV}} = \frac{1}{2} m_{\text{SUV}} v_{\text{SUV}}^{2} \) Given that the mass \( m_{\text{SUV}} = 2 \times 10^{3} \text{ kg} \) and the speed \( v_{\text{SUV}} = 25 \text{ m/s} \): \( KE_{\text{SUV}} = \frac{1}{2} \times 2,000 \text{ kg} \times (25 \text{ m/s})^{2} \) \( KE_{\text{SUV}} = \frac{1}{2} \times 2,000 \text{ kg} \times 625 \text{ m}^{2}/\text{s}^{2} \) \( KE_{\text{SUV}} = 625,000 \text{ J} \)
05

- Compare the Kinetic Energies

Compare the calculated kinetic energies of the ice piece and the SUV: \( KE_{\text{ice}} = 1,250,000 \text{ J} \) \( KE_{\text{SUV}} = 625,000 \text{ J} \) Clearly, the kinetic energy of the ice piece is higher than that of the SUV.

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Key Concepts

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

kinetic energy
Kinetic energy is the energy an object possesses due to its motion. It is given by the formula: \( KE = \frac{1}{2} m v^{2} \). This means that kinetic energy depends on both the mass (\( m \)) and the velocity (\( v \)) of an object.

Higher speeds or greater masses result in higher kinetic energy. This is why something large and slow, like an SUV, can have substantial kinetic energy, and something small but fast, like a piece of ice entering the atmosphere, can have even more.
unit conversion
Unit conversion is critical to ensure all measurements align correctly when calculating values like kinetic energy. In our exercise, we needed to convert:

  • Mass of the ice piece from grams to kilograms.
  • Speed of the SUV from kilometers per hour to meters per second.

Knowing that 1 gram = 0.001 kilograms and that 1 kilometer/hour = \( \frac{1}{3.6} \) meters/second helps streamline these conversions.
speed calculation
Calculating speed is crucial in our kinetic energy formula. For example, the ice piece's speed of 50 kilometers/second is converted directly to 50,000 meters/second.

For the SUV moving at 90 kilometers/hour, we convert this speed to meters/second using the ratio \( \frac{1km}{3.6 m/s^{- 1}} \). Therefore, 90 km/h is equivalent to 25 m/s. This conversion is essential to accurate kinetic energy calculations as the speed directly affects the formula.
mass and velocity relationship
The mass and velocity relationship in the kinetic energy formula implies that small increases in velocity significantly impact kinetic energy. Since \( KE = \frac{1}{2} m v^{2} \), doubling the velocity results in a kinetic energy increase by a factor of four.

Similarly, increasing mass directly increases kinetic energy. The exercise shows that, despite the SUV's massive weight, the ice piece's higher velocity gives it more kinetic energy, illustrating that kinetic energy depends heavily on both mass and the square of velocity.

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Most popular questions from this chapter

If a radioactive element has a half-life of 10,000 years, what fraction of it is left in a rock after 40,000 years? a. \(1 / 2\) b. \(1 / 4\) c. \(1 / 8\) d. \(1 / 16\) e. \(1 / 32\)

Video: Watch one of the available documentaries about the Apollo missions to the Moon (for example, In the Shadow of the Moon, 2008 ). Why did the United States decide to send astronauts to the Moon? Why did the Apollo program end? Are there current plans to send people to the Moon? b. The first science fiction film was the short Voyage to the Moon (Georges Méliès, 1902 ). A version with an English narration can be viewed at https://archive.org/details/ Levoyagedanslalune. A restored digitized and colorized version was released in 2011 and can be found at http: \(/ /\) vimeo.com/39275260. Where do the "Selenians" live on the Moon? In this first cinematic depiction of contact with life from outside of Earth, what do the human astronomers do to the Selenians? Contrast what the astronomers in the film find on the Moon with what the \(A\) pollo astronauts actually saw.

Earth's mean radius is \(6,371 \mathrm{km}\), and its mass is \(6.0 \times 10^{24} \mathrm{kg}\) The Moon's mean radius is \(1,738 \mathrm{km},\) and its mass is \(7.2 \times 10^{22} \mathrm{kg}\) a. Calculate Earth's average density. Show your work; do not look this value up. b. The average density of Earth's crust is \(2,600 \mathrm{kg} / \mathrm{m}^{3}\). What does this value tell you about Earth's interior? c. Compute the Moon's average density. Show your work. d. Compare the average densities of the Moon, Earth, and Earth's crust. What do these values tell you about the Moon's composition compared to that of Earth and of Earth's crust?

Citizen science: a. Go to the website for "Moon Zoo" (http://moonzoo.org), a project that lets everyone participate in the analysis of images from NASA's Lunar Reconnaissance Orbiter. Read through the FAQ, then click on "Tutorials" and select "How to Take Part." (You will need to create an account if you haven't already done so for another Zooniverse project.) In this project you count craters on the Moon, noting where there are boulders, classifying some of these features, and looking for hardware left over from exploration missions. b. Go to the website for cosmoquest (http://cosmoquest.org) and click on "Mercury Mappers." You will need to create an account for the cosmoquest projects. Click on the circled question mark under the blue check box, and read the FAQ and watch the tutorial. What is the goal of this project? Where did the data come from? Classify some images. c. Go to the website for cosmoquest (http://cosmoquest.org) and click on "Moon Mappers." As in part (b), you will need an account. Click on the circled question mark under the blue check box and read the FAQ and watch the four tutorials. What are some of the basic features? How does the angle of the sunlight and the direction of illumination affect what you see? Now classify a few craters.

Suppose an earthquake occurs on an imaginary planet. Scientists on the other side of the planet detect primary waves but not secondary waves after the quake. This suggests that a. part of the planet's interior is liquid. b. all of the planet's interior is solid. c. the planet has an iron core. d. the planet's interior consists entirely of rocky materials. e. the planet's mantle is liquid.
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