A comet having a mass of 8.38 × 10¹¹ kg strikes the Earth at a relative speed of 30 km/s. (a) Compute the kinetic energy of the comet in megatons of TNT; the detonation of 1 million tons of TNT releases 4.2 × 10¹⁵ J of energy. (b) The diameter of the crater blasted by a large explosion is proportional to the one-third power of the explosive energy released, with 1megaton of TNT producing a crater about 1 km in diameter. What is the diameter of the crater produced by the impact of the comet? (In the past, atmospheric effects produced by impacts of comets may have been the cause of mass extinctions of many species of animals and plants; it is thought by many that dinosaurs became extinct by this mechanism.)

To tackle the problem of the comet striking Earth, we need to break it down into two parts: calculating the kinetic energy of the comet and then determining the diameter of the crater produced by its impact. Let's go through each step carefully.

Kinetic Energy Calculation

The kinetic energy (KE) of an object can be calculated using the formula:

KE = 0.5 × m × v²

Where:

• m is the mass of the object (in kilograms)
• v is the velocity of the object (in meters per second)

In this case, the mass of the comet is given as:

m = 8.38 × 10¹¹ kg

The relative speed of the comet is:

v = 30 km/s = 30,000 m/s

Now, substituting these values into the kinetic energy formula:

KE = 0.5 × (8.38 × 10¹¹) × (30,000)²

Calculating the velocity squared:

(30,000 m/s)² = 900,000,000 m²/s²

Now substituting this back into the kinetic energy equation:

KE = 0.5 × (8.38 × 10¹¹) × (900,000,000)

KE = 0.5 × 8.38 × 10¹¹ × 9 × 10⁸

KE = 0.5 × 75.42 × 10¹⁹

KE = 37.71 × 10¹⁹ J

Now, to convert this energy into megatons of TNT, we use the fact that 1 megaton of TNT is equivalent to 4.2 × 10¹⁵ J.

To find the energy in megatons:

Energy in megatons = KE / (4.2 × 10¹⁵)

Energy in megatons = (37.71 × 10¹⁹) / (4.2 × 10¹⁵)

Energy in megatons ≈ 8,973 megatons of TNT

Crater Diameter Estimation

Next, we need to determine the diameter of the crater produced by the impact. The relationship between the energy released and the crater diameter is given by:

D = k × (E^1/3)

Where:

• D is the diameter of the crater
• E is the energy released in megatons of TNT
• k is a constant that relates energy to crater size

From the problem, we know that 1 megaton of TNT produces a crater about 1 km in diameter. Thus, we can set k = 1 km for 1 megaton.

Now, substituting the energy we calculated:

D = 1 × (8,973)^1/3

Calculating the cube root:

(8,973)^1/3 ≈ 20.1 km

Therefore, the diameter of the crater produced by the impact of the comet would be approximately:

D ≈ 20.1 km

Significance of Impacts

Impacts from large celestial bodies like comets can have catastrophic effects on Earth. The energy released during such events can lead to massive craters, atmospheric disturbances, and even mass extinctions, as theorized in the case of the dinosaurs. Understanding these dynamics helps scientists assess risks and prepare for potential future impacts.