The mean diameters of X and Y, two planets in the same solar system, are 7.0 103 km and 1.4 104 km, respectively. The mass of X is 0.15 times the mass of Y. The value of g on Y is 8.6 m/s2.
(a) What is the ratio of the mean density of X to that of Y (ρX / ρY)?
(b) What is the value of g on X?
m/s2
(c) The mass of Y is 6.318 1024 kg. What is the escape speed on X?
m/s
I've done a and b, but I can't figure c out. Do I use G=6.67e-11?
To answer these two questions, we need to use the formulas for density, gravitational acceleration, and escape speed. Let's break down each question step by step.
(a) The ratio of mean density of X to that of Y (ρX / ρY) can be calculated using the formula: ρ = m/V, where ρ is the density, m is the mass, and V is the volume.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius. Since the mean diameter (d) is given, we can calculate the radius (r) as r = d/2.
For planet X, the mean diameter is 7.0 * 10^3 km, which gives a radius of rX = (7.0 * 10^3) / 2 = 3.5 * 10^3 km.
For planet Y, the mean diameter is 1.4 * 10^4 km, which gives a radius of rY = (1.4 * 10^4) / 2 = 7.0 * 10^3 km.
Next, let's calculate the volumes of planets X and Y using the formula V = (4/3)πr^3:
VX = (4/3)π(3.5 * 10^3)^3
VY = (4/3)π(7.0 * 10^3)^3
Now, we can calculate the mass (m) of planet X using the given information that its mass is 0.15 times the mass of planet Y:
mY = 6.318 * 10^24 kg
mX = 0.15 * mY
Finally, we can calculate the density (ρ) of planets X and Y by dividing their masses by their respective volumes:
ρX = mX / VX
ρY = mY / VY
The ratio of the mean density of X to that of Y (ρX / ρY) is then:
ρX / ρY = (mX / VX) / (mY / VY)
(b) The formula to calculate the value of gravitational acceleration (g) is given by:
g = G * (m / r^2), where G is the gravitational constant, m is the mass, and r is the radius.
To calculate the value of g on planet X, we'll use the mass and radius of X.
(c) The escape speed is given by the formula:
v = √(2 * g * r), where v is the escape speed, g is the gravitational acceleration, and r is the radius.
To calculate the escape speed on planet X, we'll use the gravitational acceleration and radius of X.
Let's calculate each part: