Question 1125373: Hi, Please help me solve part D. Thank you.
Kepler’s Third Law of planetary motion states that the square of the period T , in years, of a
planet (the time taken for a complete revolution around the sun) is proportional to the cube of its average
distance d, in millions of km, from the sun.
(b) Find the constant of proportionality. Note that the average distance the Earth is from the sun is 150 million kilometers, according to Wikipedia. You will also need some of your Earth knowledge.
(c) Neptune is approximately 30 times as far away from the sun as Earth. Determine the period of Neptune – using the proportionalty model in this question. Does your answer correspond with the value from other sources?
(d) Suppose in the Andromeda galaxy, planet X is a certain average distance away from its Sun. Another planet, Y , is twice the average distance away from the sun as X. In percentage terms, what is the period of planet Y in relation to planet X? (You may assume that Kepler’s law applies not only in our own galaxy
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i thought i answered this before, but i couldn't find the solution on algebra.com so something may have gone wrong in posting it.
i'll try again.
direct variation equation is y = k * x
in your problem, replace y with T^2 and x with D^3 and the formula becomes T^2 = k * D^3.
T is the number of years for the planet to make one revolution around the sun and D is the distance of the planet from the sun in kilometers.
for earth, the equation becomes 1^2 = k * 150^3.
solve for k to get k = 1/150^3.
confirm with earth.
equation of T^2 = k * D^3 becomes T^2 = 1/150^3 * 150^3.
solve for T to get T = sqrt(1^2) = 1.
that should answer part B.
for part C, you have neptune is 30 times as far from the sun as the earth is.
the formula becomes T^2 = k * D^3 becomes T^2 = 1/150^3 * (30 * 150)^3 which becomes T^2 = 1/150^3 * 30^3 * 150^3 which becomes T^2 = 30^3 which becomes T^2 = 27000.
solve for T to get T = sqrt(27000) = 164.3167873.
from the web, the number of years neptune takes to make one revolution around the sun is 164.79 years.
that's pretty close to what the formula determined and is therefore your solution to part C.
part D question and answer are shown below.
(d) Suppose in the Andromeda galaxy, planet X is a certain average distance away from its Sun. Another planet, Y , is twice the average distance away from the sun as X. In percentage terms, what is the period of planet Y in relation to planet X? (You may assume that Kepler’s law applies not only in our own galaxy
let the distance from the sun for planet X be represented by x.
planet Y is twice the distance from the sun, so the distance from the sun for planet Y is equal to 2x.
the formula for planet X is T^2 = k * x^3
the formula for planet Y is T^2 = k * (2x)^3
simplify these equations to get:
the formula for planet X is T^2 = k * x^3
the formula for planet Y is T^2 = k * 8 * x^3
solve for T in each of these equations to get:
T = sqrt(k * x^3) for planet X.
T = sqrt(k * 8 * x^3) for planet Y.
these equations can also be written as:
T = sqrt(k) * sqrt(x^3) for planet X.
T = sqrt(k) * sqrt(8) * sqrt(x^3) for planet Y.
the ratio of the time it takes planet Y to the time it takes planet X to make one revolution about the sun is:
(sqrt(k) * sqrt(8) * sqrt(x^3)) / (sqrt(k) * sqrt(x^3))
simplify to get:
the ratio of the time it takes planet Y to the time it takes planet X to make one revolution about the sun is:
sqrt(8) / 1 which is equal to 2.828427125.
this means it takes planet Y 2.828427125 times as long to make one revolution about the sun as it takes for planet X.
multiply that by 100 and it becomes 282.8427125%.
this means that the amount of time it takes planet Y to make one revolution about the sun is 282.8427125% of the amount of time it takes planet X to make one revolution about the sun.
as an example:
assume the distance from the sun for planet X is 150 kilometers (sound familiar?).
since the distance of planet Y from the sun is 2x, then the distance from the sun for planet Y is 300 kilometers.
since k is equal to 1/150^3, then:
formula for planet X would be T^2 = 1/150^3 * 150^3 = 1.
solve for T to get T = sqrt(1) = 1.
formula for planet Y would be T^2 = 1/150^3 * 300^3 = 8.
solve for T to get T = sqrt(8) = 2.828427125.
it takes planet X 1 year so make one revolution around the sun.
it takes planet Y 2.828427125 years to make one revolution around the sun.
2.828427125 / 1 = 2.828427125 * 100 = 282.828427125%
hopefully this is the answer you are looking for.
let me know how we did.
|
|
|
