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Chapter 8: Problem 46

Compare the kinetic energy 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 \(\left(\text { mass }=2 \times 10^{3} \mathrm{kg}\right)\) speeding down the highway at \(90 \mathrm{km} / \mathrm{h}\)

Short Answer

Expert verified
The SUV's kinetic energy is 625000 J, while the ice's is 1250 J. The SUV has significantly more kinetic energy.

Step by step solution

01

- Convert Units

First, convert all measurements to the same units. The speed of the ice is already in meters per second (m/s), but the speed of the SUV is in kilometers per hour (km/h).Convert 90 km/h to m/s using the formula: 90 km/h = 90 * (1000 m / 1 km) * (1 hr / 3600 s) = 25 m/s.
02

- Calculate Kinetic Energy of Ice

Use the formula for kinetic energy: \( KE = \frac{1}{2}mv^2 \)For the ice, the mass (m) is 1 gram or 0.001 kg, and the speed (v) is 50,000 m/s.\( KE_{ice} = \frac{1}{2} * 0.001 kg * (50000 m/s)^2 = 1250 J \).
03

- Calculate Kinetic Energy of SUV

Again, use the kinetic energy formula:\( KE = \frac{1}{2}mv^2 \)For the SUV, the mass (m) is 2000 kg, and the speed (v) is 25 m/s.\( KE_{SUV} = \frac{1}{2} * 2000 kg * (25 m/s)^2 = 625000 J \).
04

- Compare the Results

Now, compare the kinetic energies calculated. The kinetic energy of the ice is \( 1250 J \), and the kinetic energy of the SUV is \( 625000 J \).Therefore, the kinetic energy of the SUV is significantly greater than that of the ice.

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Key Concepts

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

kinetic energy formula
The kinetic energy formula is one of the core concepts in physics. It is used to determine the energy possessed by an object due to its motion. The formula is given by: \( KE = \frac{1}{2}mv^2 \), where \( KE \) is the kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity. By applying this formula, you can calculate how much energy an object has when it's moving. The kinetic energy is always a positive value, meaning it cannot be less than zero. The mass should be in kilograms (kg) and the velocity in meters per second (m/s) for the formula to yield a result in joules (J), the SI unit for energy.
unit conversion
Unit conversion is a critical step in solving problems related to physical quantities. To compare kinetic energies accurately, all units must be consistent. In our exercise, we needed to convert the SUV's speed from kilometers per hour (km/h) to meters per second (m/s). The conversion formula used is: \( 1 \text{ km/h} = \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s} \). Using this, the speed of 90 km/h can be converted as follows: \( 90 \text{ km/h} \times \frac{5}{18} = 25 \text{ m/s} \). Consistent units are necessary for applying formulas to ensure the results are accurate and meaningful.
speed and velocity
Speed and velocity both describe how fast an object is moving, but they are not the same. Speed is a scalar quantity, meaning it only has magnitude (how fast something is moving). Velocity, on the other hand, is a vector quantity, which means it has both magnitude and direction. For kinetic energy calculations, we primarily use the magnitude of velocity. In our example, the ice had a speed of \(50 \text{ km/s}\) while the SUV had a speed of \(90 \text{ km/h}\), which converts to \(25 \text{ m/s}\). These values are then squared in the kinetic energy formula, greatly affecting the kinetic energy results.
mass and energy calculation
The mass of an object plays a crucial role in determining its kinetic energy. Larger masses result in higher kinetic energy when moving at the same speed. In our example, the mass of the ice was \(0.001 \text{ kg} \) and the SUV's mass was \(2000 \text{ kg} \). By applying the kinetic energy formula: For the ice: \( KE_{ice} = \frac{1}{2} \times 0.001 \text{ kg} \times (50000 \text{ m/s})^2 = 1250 \text{ J} \). For the SUV: \( KE_{SUV} = \frac{1}{2} \times 2000 \text{ kg} \times (25 \text{ m/s})^2 = 625000 \text{ J} \). Clearly, the SUV, despite its slower speed, has significantly more kinetic energy due to its much larger mass.
mechanics
Mechanics is the branch of physics that deals with the motion of objects and the forces acting on them. Kinetic energy is a fundamental concept in mechanics, as it quantitatively describes the energy due to motion. In our exercise, we explored how the kinetic energy of two objects (a small ice particle and a large SUV) is determined and compared. This comparison helps in understanding how different factors like mass and velocity contribute to the overall kinetic energy. Mechanics provides the tools to analyze such physical situations through principles like energy, forces, and motion.

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Most popular questions from this chapter

Earth's interior is heated by a. angular momentum and gravity. b. radioactive decay and gravity. c. radioactive decay and tidal effects. d. angular momentum and tidal effects. e. gravity and tidal effects.

The International Astronomical Union adopted a plan to have all features on Venus (with a few exceptions for large features discovered earlier) named after historical or mythological women. Categories of features are paired with specific subcategories of women; for example, large craters (greater than about \(15 \mathrm{km}\) ) are named after famous women, smaller craters have female first names, mountains are named after goddesses, and so on. Go to the Venus Crater Database (http://lpi.usra.edu/resources/vc/vchome.html), click on "Craters by Image Map," click on a region to zoom in on, and then select a few craters. For each crater, consider these questions: What is the "morphologic" class of the crater? Is there another crater very close by? What is the crater's diameter? How high is the elevation? Who was the large crater named after (if you didn't get a large one, go back to the starting address and click on "Craters by Descending Diameter")

Explain plate tectonics and identify the only planet on which this process has been observed.

Can all rocks be dated with radiometric methods? Explain.

Volcanoes have been found on all of the terrestrial planets. Where are the largest volcanoes in the inner Solar System?
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