physics wiz?

Question

the speed with which a body should be projected from the earth's surface in order to reach an infinite distance is about 1.1*10^4 m/s.Estimate the speed of escape from the moon.
[(mass of earth)/(mass of moon)=81
(radius of earth)/ (radius of moon)=3.7]

Answer 1

This is most easily treated as an energy problem, with the gravitational body at the center of a potential energy well. If the potential energy at a distance of infinity is taken to be zero, then the gravitational potential energy of a body of mass mb at a distance r from the center of a planet with mass m will be

P = -G*mb*m/r
The kinetic energy of the body will be
K = 1/2*mb*v²

We don't know mb, we don't know - more precisely, don't need - the universal gravitational constant G, the actual values for the mass of the earth or the moon, but we do know the ratios me/mm = 81 and re/rm = 3.7. We also know the escape velocity from earth. At this escape velocity, K + P = 0. Solving for the square of escape velocity from earth ve², we find that mb cancels and

ve² = 2 * G *me / re

We can get rid of the other constants we don't know or don't need by finding the escape velocity from the moon vm² in the same way and do a ratio, giving

vm²/ve² = (mm/me) / (rm/re)
= (re/rm) / (me/mm)
or
vm = ve * √[(re/rm) / (me/mm)]
= 1.1e4 * √(3.7/81)
= 2.4e3 m/s

Answer 2

For a planet/moon of mass "M" and radius "R", the escape velocity is:

V = sqrt(GM/R)

So, the equations for escape velocity for earth and moon are:

(1): V_earth = sqrt(GM_earth/R_earth)
(2): V_moon = sqrt(GM_moon/R_moon)

Next, just divide equation (1) by equation (2). This will make the "G" cancel out, and you'll be left with expressions in terms of "M_earth/M_moon" and "R_earth/R_moon", which are already given to you in the problem. Finally, just solve for V_moon.

Answer 3

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