The average distance of the planet Mercury from the sun is 0.39 times the average distance of the earth from t?
The average distance of the planet Mercury from the sun is 0.39 times the average distance of the earth from the sun. How long is a year on Mercury in units of earth years?
earth years?
.39^1.5= .24 years = 88.5 days
Let Re be the distance of the earth from the sun
and Rm the distance of mercury from the sun.
Let Ve be the orbital speed of the earth and
let Vm be the orbital speed of mercury around the sun.
The gravitational force between the sun (mass M) and earth is balanced by the centrifugal force
GMeM/Re² = MeVe²/Re
so GM/Re = Ve²
or Ve = √(GM/Re)..(i)
Similarly, for Mercury
GMmM/Rm² = MmVe²/Rm
which gives
Vm = √(GM/Rm)..(ii)
Dividing equation (ii) by eqn (i), we get
Vm/Ve = √(GM/Rm)/√(GM/Re)
Vm/Ve = √(Re/Rm)
Vm = Ve√(Re/Rm) …(iii)
The orbital distance covered in 1 year, (equals its period Te), by the earth is 2πRe
Te = 2πRe/Ve…(iv)
The period of Mercury is
Tm = 2πRm/Vm
= 2πRm/{Ve√(Re/Rm)} using equation (iii)
= 2π*0.39Re/{Ve√(1/0.39)} where we used Rm = 0.39Re
= 0.39√(0.39)* ( 2πRe/Ve)
= 0.244Te where we used equation (iv). So
Tm = 0.244Te where Te is 1 earth year.
One year on Mercury is 0.244 earth year.
Hope this helps