Problem 15 - University Physics with Modern Physics

Question

Meteor Crater. About \(50,000\) years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Measurements from 2005 estimate that this meteor had a mass of about \(1.4 \times 10^{8}\) kg (around \(150,000\) tons) and hit the ground at a speed of 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 released by a \(1.0-\) megaton nuclear bomb? (A megaton bomb releases the same amount of energy as a million tons of TNT, and 1.0 ton of TNT releases \(4.184 \times 10^{9}\) J of energy.)

Step-by-Step Solution

Answer

The meteor delivered approximately 24.1% the energy of a 1-megaton bomb.

Step 1: Identify Given Values for Kinetic Energy

To solve for the kinetic energy (KE) delivered by the meteor, we identify the given values: the mass (\(m\)) of the meteor is \(1.4 \times 10^{8} \text{ kg}\), and the velocity (\(v\)) is \(12 \text{ km/s} = 12000 \text{ m/s}\).

Step 2: Use Kinetic Energy Formula

Kinetic energy is given by the formula:\[ KE = \frac{1}{2}mv^2 \]Substitute the known values:\[ KE = \frac{1}{2} \times (1.4 \times 10^{8}) \times (12000)^2 \]

Step 3: Calculate the Kinetic Energy

Perform the calculation:\[ KE = 0.5 \times 1.4 \times 10^{8} \times 1.44 \times 10^{8} = 1.008 \times 10^{15} \text{ J}\]Thus, the kinetic energy delivered by the meteor is approximately \(1.008 \times 10^{15} \text{ Joules}\).

Step 4: Energy Released by a Megaton Bomb

Understand that a 1-megaton bomb releases energy equivalent to a million tons of TNT. Knowing that 1 ton of TNT releases \(4.184 \times 10^{9} \text{ J}\), calculate the energy from a 1-megaton bomb:\[ 1 \text{ megaton} = 10^{6} \times 4.184 \times 10^{9} \text{ J} = 4.184 \times 10^{15} \text{ J}\].

Step 5: Compare the Two Energies

Compare the kinetic energy of the meteor to the energy released by a 1-megaton bomb:The meteor's kinetic energy is \(1.008 \times 10^{15} \text{ J}\), and a 1-megaton bomb releases \(4.184 \times 10^{15} \text{ J}\).To find the ratio:\[ \frac{1.008 \times 10^{15}}{4.184 \times 10^{15}} \approx 0.241 \]The meteor delivered energy approximately 24.1% of that of a 1-megaton bomb.

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. It's a vital concept in physics because it helps us understand how energy moves and changes form. Imagine a car speeding down a highway—the faster it goes, the more kinetic energy it has.
To calculate kinetic energy (KE), we use the formula: \[ KE = \frac{1}{2}mv^2 \]where:

• \( m \) is the mass of the object, measured in kilograms (kg),
• \( v \) is the velocity, or speed, of the object in meters per second (m/s).

This formula tells us that KE increases with both the mass of the object and the square of its velocity. Hence, even a small increase in speed can significantly boost kinetic energy. When the meteor hit near Flagstaff, Arizona, its massive size and high speed resulted in a staggering amount of kinetic energy being released upon impact.

Energy Comparison

Energy comparison involves evaluating the relative amounts of energy between two or more phenomena. Understanding these comparisons can highlight just how powerful or subtle different energy releases are.
In our meteor example, comparing its kinetic energy to that of a nuclear bomb gives us a significant perspective. A 1-megaton nuclear bomb releases an enormous amount of energy, calculated as:\[ 1 \text{ megaton} = 10^{6} \text{ tons of TNT} \times 4.184 \times 10^{9} \text{ J/ton} = 4.184 \times 10^{15} \text{ J} \]The energy from the meteor impact was calculated to be approximately \(1.008 \times 10^{15} \text{ J}\). By dividing these two energies:\[ \frac{1.008 \times 10^{15}}{4.184 \times 10^{15}} \approx 0.241 \]we see that the meteor delivered about 24.1% of the energy of a 1-megaton bomb. This kind of comparison helps illustrate the immense natural forces that can be unleashed during a meteor impact.

Meteor Impact

Meteor impacts are dramatic physical phenomena that occur when a meteor, an object from space, strikes a planetary body, like Earth. These impacts can release a significant amount of energy due to the high velocity at which meteors typically travel.
When the meteor crashed approximately 50,000 years ago near present-day Flagstaff, it was moving at a speed of 12 km/s. This velocity, combined with the meteor's vast mass (estimated at \(1.4 \times 10^{8} \text{ kg}\)), resulted in the release of an enormous amount of energy.Understanding the effects of such impacts involves calculating the kinetic energy, as these colossal forces can cause significant geological and environmental changes. The crater left behind in Arizona is a stark, visible reminder of the power of meteor impacts.

Physics Calculations

Physics calculations involve the application of mathematical principles and formulas to understand and quantify the natural world. These calculations are crucial for solving physics problems by providing a structured way to find unknown values.
For instance, in calculating the kinetic energy of the meteor, we first identified the relevant information:

• The mass of the meteor: \(1.4 \times 10^{8} \text{ kg}\).
• Its velocity: 12 km/s, converted to 12000 m/s.

Then, by applying the kinetic energy formula \( KE = \frac{1}{2}mv^2 \), we performed the necessary mathematical steps:

• Substituted the known values into the formula.
• Computed \( KE = 0.5 \times 1.4 \times 10^{8} \times 1.44 \times 10^{8} \) to find \(1.008 \times 10^{15} \text{ J}\).

This logical, step-by-step approach aids students in mastering physics concepts and applying them to real-world scenarios. Understanding these calculations helps in grasping the magnitude of forces and energies at play, especially in events like a meteor impact.