Astronomers estimate that a 2.0-km-diameter asteroid collides with the Earth once every million years. The collision could pose a threat to life on Earth. Assume a spherical asteroid has a mass of 3200 kg for each cubic meter of volume and moves toward the Earth at 15 km/s. How much destructive energy could be released when it embeds itself in the Earth?

Verified step by step guidance

1. Step 1: Calculate the volume of the asteroid. Since the asteroid is spherical, use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the asteroid. The diameter is given as 2.0 km, so the radius is \( r = 1.0 \ \text{km} = 1000 \ \text{m} \). Substitute this value into the formula.
2. Step 2: Determine the mass of the asteroid. The mass is given by \( m = \rho V \), where \( \rho \) is the density of the asteroid (3200 kg/m³) and \( V \) is the volume calculated in Step 1. Multiply the density by the volume to find the mass.
3. Step 3: Calculate the kinetic energy of the asteroid. The formula for kinetic energy is \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the asteroid (from Step 2) and \( v \) is its velocity (15 km/s = 15000 m/s). Substitute these values into the formula.
4. Step 4: Simplify the expression for the kinetic energy. Perform the necessary multiplications and powers to express the energy in joules (J). This value represents the destructive energy released upon collision.
5. Step 5: Reflect on the result. The energy calculated is typically expressed in terms of megatons of TNT for comparison. To convert joules to megatons of TNT, use the conversion factor: \( 1 \ \text{megaton of TNT} = 4.184 \times 10^{15} \ \text{J} \). Divide the energy in joules by this factor to express the result in megatons of TNT.

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is the mass and v is the velocity. In the context of the asteroid, its kinetic energy at the moment of impact will determine the amount of destructive energy released upon collision with the Earth.

Volume and Density

The volume of an object is the amount of space it occupies, while density is the mass per unit volume. For the spherical asteroid, its volume can be calculated using the formula V = (4/3) * π * r^3, where r is the radius. Given the density of 3200 kg/m³, the mass can be derived from the volume, which is essential for calculating the kinetic energy.

Energy Transfer

Energy transfer refers to the process by which energy moves from one system to another. In the case of the asteroid colliding with Earth, the kinetic energy of the asteroid is transferred to the Earth upon impact, resulting in destructive energy that can cause significant damage. Understanding this concept is crucial for assessing the potential impact of such collisions.