Problem 62 - University Physics with Modern Physics

Question

A Near-Earth Asteroid. On April \(13,2029\) (Friday the 13th!), the asteroid 99942 Apophis will pass within \(18,600\) mi of the earth- about \(\frac{1}{13}\) the distance to the moon! It has a density of \(2600 \mathrm{kg} / \mathrm{m}^{3},\) can be modeled as a sphere 320 \(\mathrm{m}\) in diameter, and will be traveling at 12.6 \(\mathrm{km} / \mathrm{s}\) . (a) If, due to a small disturbance in its orbit, the asteroid were to hit the earth, how much kinetic energy would it deliver? (b) The largest nuclear bomb ever tested by the United States was the "Castle/Bravo" bomb, having a yield of 15 megatons of TNT. (A megaton of TNT releases \(4.184 \times 10^{15}\) J of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis?

Step-by-Step Solution

Answer

(a) The asteroid's kinetic energy is approximately \(5.2 \times 10^{19}\) J. (b) This energy is equivalent to about 12,430 Castle/Bravo bombs.

Step 1: Calculate the Volume of the Asteroid

First, we calculate the volume of the asteroid using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \). The diameter of the asteroid is given as 320 meters, so its radius \( r \) is 160 meters. Thus,\[V = \frac{4}{3} \pi (160)^3\]

Step 2: Calculate the Mass of the Asteroid

Next, we use the asteroid's density to find its mass. The formula for mass when density and volume are known is \( m = \text{density} \times V \). The density is given as 2600 kg/m³. Plugging the values in,\[m = 2600 \times \frac{4}{3} \pi (160)^3\]

Step 3: Calculate the Kinetic Energy of the Asteroid

The kinetic energy (KE) of an object is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass and \( v \) is the velocity. The velocity of the asteroid is 12.6 km/s, which is 12600 m/s. Substituting the known values,\[KE = \frac{1}{2} \times \text{mass} \times (12600)^2\]

Step 4: Calculate Energy Equivalent in Megatons of TNT

To compare the kinetic energy with the energy of the Castle/Bravo bomb, we need to convert it into megatons of TNT. The energy of one megaton of TNT is \(4.184 \times 10^{15}\) J. If \( E_A \) is the kinetic energy, the equivalent number of bombs is:\[\frac{E_A}{4.184 \times 10^{15}}\]

Key Concepts

Kinetic Energy Calculations

Kinetic energy is the energy an object possesses due to its motion. Calculating it is crucial for understanding the impact an asteroid might have upon hitting Earth. To find this energy, use the formula: \[ KE = \frac{1}{2} m v^2 \] Here, \( m \) represents the mass of the object, and \( v \) is its velocity. Velocity is measured in meters per second (m/s), and mass in kilograms (kg).

• First, determine the mass of the asteroid (which we'll cover in another section).
• Convert the velocity from kilometers per second (km/s) to meters per second (m/s) by multiplying it by 1,000.
• Substitute these values into the formula to find the kinetic energy in joules (J). J is the standard unit of energy.

Understanding kinetic energy helps gauge the potential damage an asteroid can cause if it crashes into Earth, as energy increases greatly with speed and mass.

Volume of a Sphere

To determine the volume of an object that can be approximated as a sphere, like an asteroid, we use the formula for the volume of a sphere:\[ V = \frac{4}{3} \pi r^3 \]This formula requires the radius \( r \) of the sphere, which is half the diameter. The diameter of asteroid 99942 Apophis is 320 meters, thus the radius is 160 meters.

• First, half the diameter to find the radius.
• Then, substitute the radius into the formula.
• Calculate the volume in cubic meters (m³).

This step is vital because the volume is used to find the asteroid's mass when coupled with its density. Volume calculation lays the groundwork for further analysis on impact effects.

Nuclear Bomb Yield

The energy yield of a nuclear bomb, like the "Castle/Bravo" bomb, offers a comparative framework to relate different energies. This bomb's yield serves as a benchmark for measuring large energy releases, such as that of an asteroid impact.

• The "Castle/Bravo" bomb had a yield of 15 megatons of TNT.
• One megaton of TNT releases \(4.184 \times 10^{15}\) joules of energy.
• To compare, we take the kinetic energy of the asteroid (calculated before) and divide it by the energy of one megaton of TNT.

By using these comparisons, we can conceptualize how many atomic bombs it would take to equal the energy of the asteroid's impact, providing a sense of scale to the potential devastation.

Energy Conversion

Converting energy from one form to another allows us to understand and compare vastly different energy scales. In this context, comparing the energy of an asteroid impact with the energy yield of a nuclear bomb requires transformation between kinetic energy (in joules) and TNT equivalents (in megatons).

• Kinetic energy of the asteroid is calculated in joules.
• We use the given conversion: 1 megaton of TNT is \(4.184 \times 10^{15}\) J.
• To express the asteroid's energy in terms easily understood, such as the number of "Castle/Bravo" equivalents, divide the asteroid's total energy by the energy equivalent of one megaton of TNT.

This method shifts an astronomical measurement into more familiar terms, illustrating how massive and potentially catastrophic an asteroid impact could be compared to known nuclear energies.