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)?
Wouldn't Keplers laws handle this nicely? You know the period for Earth, the radii for each.
Yes, Keplers laws can certainly be used to solve this problem.
Kepler's third law states that the square of the period of revolution of a planet is directly proportional to the cube of its average distance from the Sun.
(a) To calculate the period of revolution of the asteroid, we can use the following equation:
T^2 = k * R^3
Where T is the period of revolution, R is the average distance from the Sun, and k is a constant.
Given:
Mass of asteroid = 2.0 x 10^-4 times the mass of Earth
Distance of asteroid from the Sun = 2.9 times Earth's distance from the Sun
Since the mass of the asteroid is given as a fraction of the Earth's mass, we can substitute it into the equation as:
T^2 = k * (2.9 * R_Earth)^3
Now, we need to find the value of k. We can do this by substituting the known values for Earth:
T_Earth^2 = k * R_Earth^3
Dividing the first equation by the second equation, we get:
(T^2) / (T_Earth^2) = [(2.9 * R_Earth)^3] / R_Earth^3
Simplifying, we have:
(T^2) / (T_Earth^2) = (2.9^3)
Now, we can solve for T (the period of revolution of the asteroid) in terms of T_Earth (the period of revolution of Earth):
T^2 = (2.9^3) * T_Earth^2
Taking the square root of both sides:
T = √[(2.9^3) * T_Earth^2]
Now we know the period of revolution of the asteroid in terms of Earth's period (T_Earth). We can calculate it by substituting the known value of T_Earth, which is approximately 1 year (since we are asked to calculate the period in years).
(b) The ratio of the kinetic energy of the asteroid (K_asteroid) to the kinetic energy of Earth (K_Earth) can be found using the formula:
K_asteroid / K_Earth = (M_asteroid / M_Earth) * [(V_asteroid / V_Earth) ^ 2]
where M is the mass and V is the velocity of the respective objects.
Given:
Mass of asteroid = 2.0 x 10^-4 times the mass of Earth
We also need to consider that the distance of the asteroid from the Sun is 2.9 times Earth's distance from the Sun. Since the asteroid mass is given relative to Earth's mass, the ratio of the kinetic energy of the asteroid to Earth can be calculated as follows:
K_asteroid / K_Earth = (2.0 x 10^-4) * [(V_asteroid / V_Earth) ^ 2]
To find the velocity ratio, we can use the fact that the circular orbit speed (V) is inversely proportional to the square root of the distance from the Sun (R). Therefore:
V_asteroid / V_Earth = √(R_Earth / R_asteroid)
Substituting the known values:
V_asteroid / V_Earth = √(R_Earth / (2.9 * R_Earth))
V_asteroid / V_Earth = √(1 / 2.9)
Simplifying, we have:
V_asteroid / V_Earth = √(10 / 29)
Calculating the value of this ratio gives the answer for part (b).
Please note that the actual calculations are not included here, but you can substitute the given values into the equations to obtain the numerical answers.
Yes, Kepler's laws can indeed be used to solve this problem. Specifically, we can use Kepler's third law to find the period of revolution of the asteroid, and then use the kinetic energy equation to calculate the ratio of the asteroid's kinetic energy to Earth's.
Kepler's third law relates the period of revolution (T) of a planet or asteroid around the Sun to its average distance from the Sun (r). The equation is as follows:
T^2 = (4π^2 / GM) * r^3
Where G is the universal gravitational constant (6.67 * 10^-11 N m^2/kg^2) and M is the mass of the Sun (1.98 * 10^30 kg).
Let's go step by step to solve this problem:
(a) Calculate the period of revolution of the asteroid in years:
Step 1: Calculate the mass of the asteroid (M_asteroid):
The mass of the asteroid is given as 2.0 * 10^-4 times the mass of Earth (M_earth). So, we can calculate it as follows:
M_asteroid = (2.0 * 10^-4) * M_earth
Step 2: Calculate the average distance of the asteroid from the Sun (r_asteroid):
The distance of the asteroid from the Sun is 2.9 times Earth's distance from the Sun (r_earth). So, we can calculate it as follows:
r_asteroid = 2.9 * r_earth
Step 3: Substitute the values into Kepler's third law equation:
T_asteroid^2 = (4π^2 / GM) * r_asteroid^3
Now, we need to convert the units of T_asteroid from seconds to years. 1 year = 3.154 * 10^7 seconds.
(b) Calculate the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth (K_asteroid / K_earth):
The kinetic energy of a planet or asteroid is given by the equation:
K = (1/2) * M * V^2
Where M is the mass of the object and V is its orbital velocity.
Step 1: Calculate the orbital velocity of the asteroid (V_asteroid):
Using the circular motion equation for the gravitational force, we can relate the orbital velocity to the distance of the asteroid from the Sun (r_asteroid) and the mass of the Sun (M_sun) as follows:
V_asteroid = √(GM_sun / r_asteroid)
Step 2: Calculate the kinetic energy of the asteroid (K_asteroid):
K_asteroid = (1/2) * M_asteroid * V_asteroid^2
Step 3: Calculate the kinetic energy of Earth (K_earth):
K_earth = (1/2) * M_earth * V_earth^2
Finally, we can find the ratio of K_asteroid to K_earth by dividing K_asteroid by K_earth:
Ratio = K_asteroid / K_earth
By following these steps and plugging in the appropriate values, you should be able to calculate the period of revolution of the asteroid and the ratio of the kinetic energy of the asteroid to Earth's kinetic energy.