A person’s weight, w, on a planet of radius r is given by: w= kr^-2 k < 0 where the constant, k, depends on the masses of the person and the planet.What fraction of the Earth’s radius must an equally massive planet have for a person to weigh ten times more on that planet than on Earth?
you want R where
k/R^2 = 10(k/r^2)
R^2/r^2 = 1/10
R/r = 1/√10
To find the fraction of Earth's radius at which a person would weigh ten times more on another planet, we can start by equating the weight on Earth (w_earth) to the weight on the other planet (w_other). Let's call the fraction of Earth's radius as a (a is between 0 and 1, where 1 represents Earth's radius).
Given that the weight on Earth is w_earth, we have:
w_earth = k (1^-2) -- Equation (1)
Since the person is equally massive on both planets, the weight on the other planet, w_other, will be:
w_other = k (a^-2) -- Equation (2)
We are given that the weight on the other planet is ten times more than the weight on Earth. So, we can write:
w_other = 10 * w_earth
Substituting the values from Equations (1) and (2), we get:
10 * k (1^-2) = k (a^-2)
Simplifying further:
10 = a^-2
Taking the square root of both sides:
√10 = a^(-2/2)
√10 = a^(-1)
Taking the reciprocal of both sides:
1/√10 = 1/a
Rearranging, we find:
a = √10
Therefore, the person would need to be located at a point on the other planet that is √10 times the radius of Earth to weigh ten times more on that planet.
So, the fraction of Earth's radius that the other planet must have is √10 (approximately 3.16).