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Chapter 7: Problem 20

\(\cdot\) Meteor crater. About \(50,000\) years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Recent \((2005)\) measurements estimate that this meteor had a mass of about \(1.4 \times 10^{8} \mathrm{kg}\) (around \(150,000\) tons) and hit the ground at 12 \(\mathrm{km} / \mathrm{s}\) . (a) How much kinetic energy did this meteor deliver to the ground? (b) How does this energy compare to the energy produced in one day by a standard coal-fired power plant, which generates about 1 billion joules per second?

Short Answer

Expert verified
(a) Kinetic energy: \(1.008 \times 10^{16}\) joules; (b) Meteor's energy is much greater than a day's energy from a power plant.

Step by step solution

01

Identify Known Values

From the problem, we know the following values:- Mass \( m = 1.4 \times 10^8 \) kg - Velocity \( v = 12 \, \mathrm{km/s} = 12,000 \, \mathrm{m/s} \)These values will be used in the kinetic energy formula.
02

Use the Kinetic Energy Formula

The formula for kinetic energy \( KE \) is given by:\[KE = \frac{1}{2}mv^2\]Substitute \( m = 1.4 \times 10^8 \) kg and \( v = 12,000 \) m/s into the formula.
03

Calculate Kinetic Energy

Plug the known values into the kinetic energy formula:\[KE = \frac{1}{2} \times 1.4 \times 10^8 \, \mathrm{kg} \times (12,000 \, \mathrm{m/s})^2\]Simplify the expression:\[KE = 0.7 \times 10^8 \times 144 \times 10^6 \]Continuing with calculations:\[KE = 100.8 \times 10^{14} \text{ Joules}\]This simplifies to:\[KE = 1.008 \times 10^{16} \text{ Joules}\]
04

Compare to Daily Energy from a Power Plant

A standard coal-fired power plant produces 1 billion joules per second, which is equivalent to:\[1 imes 10^9 \, \text{Joules} / \text{second}\]To find the daily energy output, multiply by the number of seconds in a day (86,400 seconds):\[ ext{Daily energy} = 1 \times 10^9 \, \text{Joules/second} \times 86,400 \, \text{seconds}\]which equals:\[8.64 \times 10^{13} \, \text{Joules}\]Compare this to the meteor's energy:\[KE = 1.008 \times 10^{16} \text{ Joules}\]
05

Conclusion on Energy Comparison

The energy delivered by the meteor \(1.008 \times 10^{16}\) joules is significantly larger than the energy produced by a power plant in a day \(8.64 \times 10^{13}\) joules.

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Key Concepts

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

Meteor Impact
Meteor impacts are extraordinary events that can cause significant changes to the Earth's surface. Imagine a colossal rock from space traveling at an incredibly high speed and crashing into our planet.
50,000 years ago, near Flagstaff, Arizona, such an event occurred. The meteor was estimated to have a mass of around 140 million kilograms and was traveling at an impressive speed of 12 kilometers per second.
This high-speed impact resulted in a massive release of energy, enough to create a large crater and drastically alter the landscape around the impact site. This incident not only emphasizes the destructive power of meteors but also their potential to affect our environment.
Meteor impacts can lead to the release of large amounts of dust and gases into the atmosphere, which might even alter the climate.
Understanding meteor impacts allows us to appreciate the dynamic nature of our planet and the role these cosmic events play in shaping Earth's geological history.
Energy Comparison
Energy comparison helps us understand the scale of energy released in different processes or events. The kinetic energy from the meteor impact in Arizona is compared to the energy output of a standard coal-fired power plant.
Let's break it down:
  • Kinetic energy from the meteor is calculated to be approximately \(1.008 \times 10^{16} \) Joules.
  • A standard power plant generates 1 billion Joules every second or \(1 \times 10^9 \) Joules per second.
  • In a day, this power plant produces \(8.64 \times 10^{13} \) Joules.
When we compare these numbers, the energy from the meteor is many times greater than the daily energy output of the power plant.
This comparison showcases the vast amount of energy released by such a space object impact. It's an excellent example of the immense forces present in our universe and how they can occasionally interact with Earth in dramatic ways.
Physics Problem Solving
Physics problem solving involves using scientific principles and formulas to understand and solve real-world phenomena. When approaching a problem like calculating the energy of a meteor impact, we follow a systematic process.
Here's a step-by-step approach:
  • Identify Known Values: Recognize the given information, such as mass and velocity for the meteor.
  • Apply Relevant Formulas: Use the kinetic energy formula \( KE = \frac{1}{2} mv^2 \) to find the energy involved.
  • Perform Calculations: Substitute the known values into the equation and perform the necessary operations to arrive at the answer.
  • Analyze and Compare: Interpret the result within the given context, such as comparing it to other energy outputs like those from a power plant.
Understanding each component in this process allows students to apply their physics knowledge to various scenarios effectively. It also enhances their problem-solving skills by giving them the tools to tackle both theoretical and practical problems in physics.

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Most popular questions from this chapter

\(\bullet\) \(\bullet\) At 7.35 cents per kilowatt-hour, (a) what does it cost to operate a 10.0 hp motor for 8.00 hr? (b) What does it cost to leave a 75 W light burning 24 hours a day?

\(\bullet\) \(\bullet\) A 68 kg skier approaches the foot of a hill with a speed of 15 \(\mathrm{m} / \mathrm{s} .\) The surface of this hill slopes up at \(40.0^{\circ}\) above the horizontal and has coefficients of static and kinetic friction of 0.75 and \(0.25,\) respectively, with the skis. (a) Use energy con- servation to find the maximum height above the foot of the hill that the skier will reach. (b) Will the skier remain at rest once she stops, or will she begin to slide down the hill? Prove your answer.

\(\bullet\) \(\bullet\) \(\mathrm{A} 25 \mathrm{kg}\) child plays on a swing having support ropes that are 2.20 \(\mathrm{m}\) long. A friend pulls her back until the ropes are \(42^{\circ}\) from the vertical and releases her from rest. (a) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? (b) How fast will she be moving at the bottom of the swing? (c) How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

\(\bullet\) \(\bullet\) \(\bullet\) Automobile air-bag safety. An automobile air bag cush- ions the force on the driver in a head-on collision by absorbing her energy before she hits the steering wheel. Such a bag can be modeled as an elastic force, similar to that produced by a spring. (a) Use energy conser- vation to show that the effec- tive force constant \(k\) of the air bag is \(k=m v_{0}^{2} / x_{\text { max }}^{2},\) where \(m\) is the mass of the driver, \(v_{0}\) is the speed of the car (and driver) at the instant of the accident, and \(x_{\max }\) is the maxi- mum distance the bag gets compressed, which, in a severe accident, would be the dis- tance from the driver's body to the steering wheel. (b) Show that the maximum force the air bag would exert on the driver is \(F_{\max }=m v_{0}^{2} / x_{\max }\) (c) Now let's put in some realistic numbers. Experimental tests have shown that injury occurs when a force density greater than \(5.0 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) acts on human tissue. (The total force is this force density times the area over which it acts. As the accom- panying figure shows, the force of the air bag acts mostly on the upper front half of the driver's body, over an area of about 2500 \(\mathrm{cm}^{2} .\) (Check your own body to see if this is reasonable.) Use this value to calculate the total force on the driver's body at the threshold of injury. (d) Use your results to calculate the effective force constant \(k\) of the air bag and the maximum speed for which the bag will prevent injury to a 65 kg driver if she is 30 \(\mathrm{cm}\) from the steering wheel at the instant of impact. Express the speed in \(\mathrm{m} / \mathrm{s}\) and mph. (e) How could you design a safer air bag for higher speed collisions? What things could you alter to do this? Would it be safe to make a stiffer air bag by inflating it more? Explain your reasoning.

\(\cdot\) A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of \(15.0^{\circ}\) with the forward direction of motion. If the tension in the rope is \(180 \mathrm{N},\) how much work does the rope do on the skier during a forward displacement of 300.0 \(\mathrm{m} ?\)
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