mass of planet is half that of earth and radius of planet is one fourth that of earth if we plan to send an artificial satellite from planet the escape velocity will be in km/s
force of gravity = G M m/R^2
if mass is m, what is escape speed from R, M
(1/2) m V^2 speed V at R = work done falling from infinity to R
(1/2) m V^2 = integral oo to R of F dr
(1/2) m V^2 = (G M m/R^2) dr
= (GM)(1/R)
V^2 = 2 G M/R
if R = 1/4 Radius earth
and mass = (1/2) mass earth
Vplanet^2 = 2 G .5 Mearth/(.25)^2 Rearth
so
Vplanet^2 = 8 times Vearth^2
Vplanet = 2.83 * Vearth
To find the escape velocity of the artificial satellite from the planet, we need to use the formula for escape velocity:
Ve = √(2 * G * Mp / Rp)
Where:
- Ve is the escape velocity,
- G is the gravitational constant (approximately 6.67430 × 10^-11 m³ kg⁻¹ s⁻²),
- Mp is the mass of the planet, and
- Rp is the radius of the planet.
Given that the mass of the planet is half that of Earth and the radius of the planet is one fourth that of Earth, we can substitute these values into the formula.
Let's assume the mass of Earth is Me and the radius of Earth is Re.
Mp = Me / 2
Rp = Re / 4
Substituting these values into the formula:
Ve = √(2 * G * (Me / 2) / (Re / 4))
Now, Ve is in m/s. To convert it into km/s, we divide it by 1000:
Ve (km/s) = Ve (m/s) / 1000
So, using the above equation, we can calculate the escape velocity of the artificial satellite from the planet.