An asteroid, whose mass is 2.0 10-4 times the mass of Earth, revolves in a circular orbit around the Sun at a distance that is 2.9 times Earth's distance from the Sun.
(a) Calculate the period of revolution of the asteroid in years.
(b) What is the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth (Kasteroid / KEarth)?
I repeat: Is there not a Kepler law that relates radius of orbit to period?
since T^2 is proportional to r^3, (a) should be sqrt(2.9^3)years.
(b) find velocity of both the earth (make radius of earth =1r and radius of asteroid=2.9r)the r's will cancel out
Yes, you are correct. Kepler's third law states that the square of the period of revolution is proportional to the cube of the semi-major axis of the orbit. We can use this law to calculate the period of revolution of the asteroid.
(a) Let's assume that the period of revolution of Earth is 1 year. Since the asteroid is at a distance that is 2.9 times Earth's distance from the Sun, its semi-major axis is 2.9 times the semi-major axis of Earth's orbit.
Using the proportionality relationship, we have:
(T_asteroid^2 / T_earth^2) = (r_asteroid^3 / r_earth^3)
Plugging in the values, we have:
(T_asteroid^2 / 1^2) = (2.9^3 / 1^3)
Simplifying the equation, we get:
T_asteroid^2 = 2.9^3
Taking the square root of both sides, we find:
T_asteroid = sqrt(2.9^3) years
So, the period of revolution of the asteroid is approximately sqrt(2.9^3) years.
(b) To find the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth, we need to compare their velocities.
The velocity of an object in circular orbit can be calculated using the equation: v = (2πr) / T, where v is the velocity, r is the radius of the orbit, and T is the period of revolution.
Let's assume the radius of Earth's orbit is 1r and the radius of the asteroid's orbit is 2.9r.
The velocity of Earth is:
v_earth = (2π * 1r) / 1 year
The velocity of the asteroid is:
v_asteroid = (2π * 2.9r) / sqrt(2.9^3) years
The ratio of the kinetic energies can be calculated using the equation: K_asteroid / K_earth = (m_asteroid * v_asteroid^2) / (m_earth * v_earth^2)
Substituting the values, we get:
K_asteroid / K_earth = [(2.0 * 10^-4) * (2π * 2.9r) / sqrt(2.9^3) years]^2 / [(1) * (2π * 1r) / 1 year]^2
Simplifying the equation, we find:
K_asteroid / K_earth = [(2.0 * 10^-4) * (2.9)^2 / sqrt(2.9^3)]^2
By plugging in the values, you can calculate the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth.