One of Kepler's three laws of planetary motion states that the square of the period, P, of a body orbiting the sun is proportional to the cube of its average distance, d, from the sun. The Earth has a period of 365 days and its distance from the sun is approximately 93,000,000 miles.
a.) Find P as a function of d.
P(d)=
P = k d^3
given, when P = 365 days, d = 93 000 000 miles
365 = k(93000000)^3
solve for k, you will need scientific notation, then rewrite the equation
Let me know what you get for k
I got 4.5378*10^-22 for k.
correct
Still incorrect according to WebWork
I assumed you had Kepler's law stated correctly, then looked it up and found:
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
so it would be
P^2 = k d^3
try it with this change
To find P as a function of d, we need to express the relationship between P and d using Kepler's third law.
According to Kepler's third law, the square of the period, P, is proportional to the cube of the average distance, d, from the sun. Mathematically, this relationship can be written as:
P^2 ∝ d^3
To remove the proportionality symbol and find the exact relationship, we need to introduce a constant of proportionality, which we'll call k:
P^2 = k * d^3
Now we need to find the value of k. We can do this by using the values of P and d for a known planet, such as Earth.
Given:
P (period of Earth) = 365 days
d (average distance of Earth from the sun) = 93,000,000 miles
Substituting these values into the equation, we get:
(365)^2 = k * (93,000,000)^3
Solving for k:
k = (365)^2 / (93,000,000)^3
Now that we have the value of k, we can express P as a function of d:
P(d) = sqrt(k * d^3)
Substituting the value of k we found:
P(d) = sqrt((365)^2 / (93,000,000)^3) * d^(3/2)
So the function P(d) is given by:
P(d) = sqrt((365)^2 / (93,000,000)^3) * d^(3/2)