Consider an asteroid of radius 2 km and density 2,500 kgm3 impacting the Earth. When it collides with the surface its kinetic energy is, to a large extent, dumped into the environment, some as heat vaporizing rock and some as mechanical energy throwing rock to great distance and height. If the asteroid were released from rest a great distance from Earth (you can assume this is infinite) find its kinetic energy at impact. Express your result in megatons. These are units often used to measure nuclear weapon energy yields; a megaton is the energy released by one million tons of TNT and equals 4.2×1015 Joules. Round your answer to two significant digits.

Question

Consider an asteroid of radius 2 km and density 2,500 kgm3 impacting the Earth. When it collides with the surface its kinetic energy is, to a large extent, dumped into the environment, some as heat vaporizing rock and some as mechanical energy throwing rock to great distance and height. If the asteroid were released from rest a great distance from Earth (you can assume this is infinite) find its kinetic energy at impact. Express your result in megatons. These are units often used to measure nuclear weapon energy yields; a megaton is the energy released by one million tons of TNT and equals 4.2×1015 Joules. Round your answer to two significant digits.

Answer 1

First compute the mass of the asteroid from its density and radius,r. Call it m.

m = (4/3)pi*r^3

Then compute E = G*M*m/R, where
R is the radis of the Earth
M is the mass of the Earth
m is the mass of the asteroid
G is Newton's universal constant of gravity

E is the energy of the collision,in Joules, assuming the asteroid approaches from infinity. Convert it to megatons of TNT using the specified conversion factor.

Answer 2

To find the kinetic energy of the asteroid at impact, we can use the following formula:

Kinetic Energy (KE) = (1/2) * mass * velocity^2

First, we need to find the mass of the asteroid using its density and volume. The formula for the volume of a sphere is:

Volume = (4/3) * π * radius^3

Substituting the given values:

Volume = (4/3) * π * (2 km)^3

Now let's convert the radius to meters to keep the units consistent:

Volume = (4/3) * π * (2,000 m)^3

Next, we need to find the mass by multiplying the volume with the density:

Mass = Volume * Density

Mass = (4/3) * π * (2,000 m)^3 * 2,500 kg/m^3

Now, we can calculate the kinetic energy. However, to determine the velocity, we need to consider the fact that the asteroid is released from a great distance from Earth (essentially infinite). In this case, the potential energy is converted entirely into kinetic energy.

Potential Energy (PE) = Kinetic Energy (KE)

Gravitational Potential Energy (PE) = -G * (mass_earth * mass_asteroid) / distance

Since the distance is infinite, the potential energy becomes zero:

PE = 0

KE = 0

This means that the initial kinetic energy of the asteroid is zero since it is released from rest.

So, the kinetic energy at impact is zero megatons (0 MT).