1. Consider an asteroid of radius 2 km and density 2,500 kg/m^3 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 is released from rest a great distance from Earth (you can assume this is infinite) and falls under Earth's gravity, find its kinetic energy at impact.
Express your result in megatons.
2. Consider the same asteroid initially in the asteroid belt at a distance 3 AU from the Sun and 2 AU from Earth. As it falls towards Earth, it acquires kinetic energy (neglect its initial motion) both by being accelerated towards the Sun and towards the Earth. In Question 9 we neglected the effect of Solar gravity. Now, ignoring Earth's gravity, find the change in the potential energy of the asteroid due to the Sun’s gravitational force, in megatons.
Erm...I don't think we're going to solve all these problems for you you need to at least make some effort
To find the kinetic energy of the asteroid at impact, we can use the equation for kinetic energy:
Kinetic Energy = (1/2) * mass * velocity^2
However, to find the velocity of the asteroid at impact, we need to consider its potential energy at its initial position (infinite distance from Earth). Since the asteroid is released from rest, its initial kinetic energy is zero.
The potential energy at an infinite distance from Earth is also zero, so we can set the initial potential energy to zero and use the principle of conservation of energy to find the final kinetic energy at impact.
Now, let's solve for the kinetic energy at impact:
1. First, we need to find the mass of the asteroid.
Mass = (4/3) * pi * radius^3 * density
Radius = 2 km = 2000 m
Density = 2500 kg/m^3
Mass = (4/3) * 3.1415 * (2000)^3 * 2500
2. Next, we need to find the velocity of the asteroid at impact using conservation of energy.
The initial potential energy is zero, so the final kinetic energy will be equal to the change in potential energy.
Final kinetic energy = Change in potential energy
Change in potential energy = -G * (mass_of_asteroid * mass_of_earth) / distance_from_earth
G = gravitational constant = 6.67430 × 10^-11 m^3/kg/s^2
mass_of_earth = 5.972 × 10^24 kg
distance_from_earth = radius_of_earth = 6.371 × 10^6 m (approximate radius)
Change in potential energy = - (6.67430 × 10^-11) * (mass_of_asteroid) * (mass_of_earth) / (radius_of_earth)
3. Now, we can equate the final kinetic energy to the change in potential energy.
Final kinetic energy = Change in potential energy
(1/2) * mass_of_asteroid * velocity^2 = Change in potential energy
velocity^2 = (Change in potential energy) / ((1/2) * mass_of_asteroid)
4. Finally, we can calculate the velocity and the kinetic energy at impact.
Find the square root of the velocity^2 to get the actual velocity.
Plug the velocity into the kinetic energy equation to find the kinetic energy.
velocity = sqrt((Change in potential energy) / ((1/2) * mass_of_asteroid))
kinetic energy = (1/2) * mass_of_asteroid * velocity^2
Now, let's calculate the values for the given asteroid:
1. Calculating the mass of the asteroid:
Mass = (4/3) * 3.1415 * (2000)^3 * 2500
Mass = 3.3515545166 × 10^16 kg (approximate value)
2. Calculating the change in potential energy:
G = 6.67430 × 10^-11 m^3/kg/s^2
mass_of_earth = 5.972 × 10^24 kg
distance_from_earth = radius_of_earth = 6.371 × 10^6 m
Change in potential energy = - (6.67430 × 10^-11) * (3.3515545166 × 10^16) * (5.972 × 10^24) / (6.371 × 10^6)
Change in potential energy = - 2.4882383098 × 10^26 kg * m^2 / s^2
3. Calculating the velocity and the kinetic energy:
velocity = sqrt((- 2.4882383098 × 10^26) / ((1/2) * 3.3515545166 × 10^16))
velocity = 2.9119875282 × 10^3 m/s (approximate value)
kinetic energy = (1/2) * (3.3515545166 × 10^16) * (2.9119875282 × 10^3)^2
kinetic energy = 8.8326338985 × 10^22 J (approximate value)
Now, to express the result in megatons:
1 megaton = 4.184 × 10^15 J
Dividing the kinetic energy by 4.184 × 10^15 J, we get:
kinetic energy in megatons = (8.8326338985 × 10^22) / (4.184 × 10^15)
kinetic energy in megatons = 2.111 × 10^7 megatons (approximate value)
Therefore, the kinetic energy of the asteroid at impact is approximately 2.111 × 10^7 megatons.