(a) show that a projectile with mass m can \"escape\" from the surface of a plan

Question

(a) show that a projectile with mass m can "escape" from the surface of a planet if it is launched vertically upward with a  kinetic energy greater tha mgRp where g is the acceleration due to gravity at the planet's surface and Rp is the planet's radius. Ignore the air resistance.

(b) If the planet in question is the earth at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 g/mol) equal to that required to escape? What about a hydrogen molecule (molar mass 2.02 g/mol)

(c) Repeat part (b) for the moon for which g=1.63 m/s^2 and Rp=1740 km.

(d) While the earth and the moon have similar average surface temperature, the moon has essentially no atmosphere, use your answers from part b and part c to explain why.

Explanation / Answer

a)Potential energy of the mass on the surface of the earth = GmM/Rp

This energy needs to be provided as kinetic energy of the projecticle

GmM/Rp = Rp*m * GM/Rp^2

= m*Rp *g

b) So the kinetic energy = .5 * m*v^2 = m*Rp*g

.5*(3*R*T/M) = Rp*g

find T = Rp*g*28/(1.5*R)

for hydrogen use

T = Rp*g*2.02/(1.5*R)

c)
Put the value for moon which is given in question

d)Becasue the gas molecules have enough kinetic energy to escape the surface of the moon.

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