Question

Europa orbits Jupiter at an average distance of 6.71 multiply.gif 105 km with an orbital period of 0.00972 yr. The Moon, which is one of the satellites of the Earth orbits its parent at an average distance of 3.84 multiply.gif 105 km with an orbital period of 0.07481 yr. (a) Use the above information to find the orbital speeds of Europa around Jupiter and of the Moon around the Earth. vEuropa = m/s vMoon = m/s (b) What is the expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite and the gravitational constant G? (Do not substitute numerical values; use variables only.) (c) Now use your answers from parts (a) and (b) to find the ratio of the mass of the Earth to that of Jupiter.

Answer 1

Answer:

13744.6016 m/s

1021.99194 m/s

$\frac{M_e}{M_j}=316.05354$

Explanation:

R = Radius of orbit

T = Orbital period

G = Gravitational Constant

M = Mass of the planet

Orbital velocity

$v=\frac{2\pi r}{T}\\\Rightarrow v=\frac{2\pi \times 6.71\times 10^5\times 10^3}{0.00972\times 365.25\times 24\times 3600}\\\Rightarrow v=13744.6016\ m/s$

Orbital velocity of Europa is 13744.6016 m/s

$v=\frac{2\pi r}{T}\\\Rightarrow v=\frac{2\pi \times 3.84\times 10^5\times 10^3}{0.07481\times 365.25\times 24\times 3600}\\\Rightarrow v=1021.99194\ m/s$

Orbital velocity of moon is 1021.99194 m/s

When an object is in orbit the centripetal acceleration and the gravitational acceleration balance out

$\frac{v^2}{r}=\frac{GM}{r^2}\\\Rightarrow M=\frac{v^2r}{G}$

The expression is $M=\frac{v^2r}{G}$

Mass ratio

$\frac{M_e}{M_j}=\frac{v_e^2r_e}{v_j^2r_j}\\\Rightarrow \frac{M_e}{M_j}=\frac{13744.6016^2\times 6.71\times 10^5\times 10^3}{1021.99194^2\times 3.84\times 10^5\times 10^3}\\\Rightarrow \frac{M_e}{M_j}=316.05354$

The mass ratio is $\frac{M_e}{M_j}=316.05354$