/*! 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 18 A typical meteor that hits the e... [FREE SOLUTION] | 91Ó°ÊÓ

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A typical meteor that hits the earth's upper atmosphere has a mass of only 2.5 g, about the same as a penny, but it is moving at an impressive 40 \(\mathrm{km} / \mathrm{s} .\) As the meteor slows, the resulting thermal energy makes a glowing streak across the sky, a shooting star. The small mass packs a surprising punch. At what speed would a \(900 \mathrm{kg}\) compact car need to move to have the same kinetic energy?

Short Answer

Expert verified
The compact car would need to move at a speed of approximately \(0.0014907 \, \mathrm{km/s}\) to have the same kinetic energy as the meteor.

Step by step solution

01

Calculate the kinetic energy of the meteor

To start, the mass of the meteor must be given in kg to match the SI Units. Therefore, the mass is converted from g to kg by dividing by 1000 to get \( m = 2.5/1000 = 0.0025 \, \mathrm{kg} \). The kinetic energy of the meteor can then be calculated using \( KE = 1/2 m v^2 \), which gives \( KE_{\mathrm{meteor}} = 1/2 \times 0.0025 \, \mathrm{kg} \times (40 \, \mathrm{km/s})^2 = 2000 \, \mathrm{J} \).
02

Calculate the speed of the car

We can equate and rearrange the kinetic energy equation for the car with the kinetic energy we found for the meteor: \( KE_{\mathrm{car}} = KE_{\mathrm{meteor}} = 1/2 m_{\mathrm{car}} v_{\mathrm{car}}^2 \). Re-writing for the speed of the car gives \( v_{\mathrm{car}} = \sqrt{(2 KE_{\mathrm{car}}) / m_{\mathrm{car}}} \). Substituting \( KE_{\mathrm{car}} = 2000 \, \mathrm{J} \) and \( m_{\mathrm{car}} = 900 \, \mathrm{kg} \) into the equation and simplifying gives \( v_{\mathrm{car}} = \sqrt{(2* 2000 \, \mathrm{J}) / 900 \, \mathrm{kg}} = 1.4907 \, \mathrm{m/s}\). To convert the speed into kilometer per second, divide by 1000 which gives \( v_{\mathrm{car}} = 1.4907 / 1000 = 0.0014907 \, \mathrm{km/s} \).

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Key Concepts

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

Meteor Physics
A meteor, often referred to as a shooting star, is a small piece of space debris that enters Earth's atmosphere. Despite its small size, a meteor can move at incredibly high velocities, often reaching speeds of 40 kilometers per second or more.
When a meteor travels at such high speed and encounters the Earth’s atmosphere, it experiences atmospheric friction. This friction generates an immense amount of heat, causing the meteor to burn up and produce a bright streak of light across the sky.
It's fascinating to consider the energy a tiny meteor possesses, largely due to its tremendous speed. In physics, the kinetic energy of an object is influenced both by its mass and the square of its velocity. Hence, even a miniscule mass can have a significant impact if moving quickly enough. This principle underlies why a small meteor can pack a powerful punch upon impact due to its high kinetic energy.
Energy Conversion
The phenomenon of a meteor exhibiting a bright glow as it enters Earth's atmosphere is an example of energy conversion. Initially, the meteor has high kinetic energy due to its motion through space.
As the meteor descends through the atmosphere, kinetic energy is transformed predominantly into thermal energy due to friction. This process is what produces the visible light, often seen as a shooting star.
The kinetic energy of an object can be calculated using the formula: \( KE = \frac{1}{2} mv^2 \), where \( m \) is mass and \( v \) is velocity. For the meteor, the conversion of kinetic energy into heat and light is efficient enough to be perceptible from the ground as a brilliant flash. This shows how energy can convert from one form to another, yet the total energy remains constant within the system, a key principle of physics.
SI Units Conversion
Physics formulas often require consistency in measurement units. The International System of Units (SI) is the standard system used globally. For calculations involving kinetic energy, ensuring that mass is in kilograms and velocity in meters per second is crucial.
In this exercise, we converted the meteor’s initial mass from grams to kilograms (by dividing by 1000), making it suitable for direct substitution into the kinetic energy formula. Similarly, velocities in this system are typically calculated in meters per second. The conversion from kilometers per second to meters per second involves multiplying by 1000.
This systematic approach to units allows accurate calculations and comparisons across different scenarios, such as determining the speed necessary for a car to have the same kinetic energy as the meteor.
  • Grams to kilograms: Divide by 1000.
  • Kilometers to meters: Multiply by 1000.
This ensures that all variables align, preventing errors and providing consistency in scientific calculations.

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Most popular questions from this chapter

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