SECOND SET SOLUTION 
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1)  Problem about the comet: 

a) The answer is directly given by the meaning of Kepler's second
law. 
   In fact each object moving in elliptical orbit around the Sun will 
   have the greatest speed at the perihelion, that is the closest
point 
   to the Sun.

b) Considering that the Sun is at one of the ellipse's foci, the 
   semi-major axis is given by half the sum of the distance at the 
   perihelion plus the distance at the aphelion: 

   a = 1/2 * (3.0*10^8 + 7.5*10^9) Km = 3.9*10^9 km 

   The period of the orbit is obtained using Kepler's third law. 
   In this case, it is necessary to convert Km into AU (astronomical 
   units) before using the simplified formula: 

   P^2 = k * a^3, with k = 1
   a = 3.9*10^9 km = 26.07 AU (considering that 1 AU = 1.496 * 10^8
km) 

   P = a^(3/2) = (26.07)^(3/2) years = 133.11 years 


2) Problem about your weight at the sea level and the one on Mount
Everest. 
   In this problem it was important to recognize that the
gravitational 
   force and the force of gravity (weight W) were the same. In this
case: 

   weight at the sea level = W_s = G * (M_E * m)/ R_E^2      and 
   weight on Mount Everest = W_m = G * (M_E * m)/ (R_E+h)^2. 

   Where G is the universal gravitational constant, M_E is the mass 
   of the Earth, R_E is the Earth radius, m is your mass, and h is the
]
   altitude of Mount Everest. 
   We are interested in calculating the ratio of these two weight: 
   W_s / W_m. 
   In this way we can write: 

   W_s / W_m = [G * (M_E * m)/ R_E^2] / [G * (M_E * m)/ (R_E+h)^2] =  
             = [1/ R_E^2] / [1/ (R_E+h)^2] = 
             = (R_E+h)^2 / R_E^2

   W_s / W_m = (6378 + 9.114) / 6378 = 1.00143 

   If your weight at the sea level is 140 lb, then your weight on 
   Mount Everest would be: 

   W_m = W_s / 1.00143 = 139.8 lb 


3) Problem about Jupiter's mass: 

   The formula to use is: P^2 = [4 pi^2 / G * (M_j + M_c)] * a^3
   But Callisto's mass is much smaller than Jupiter's mass, therefore:


   M_c << M_J and  P^2 = [4 pi^2 / (G * M_j)] * a^3    and 

   M_J = [4 pi^2 / (G * P^2)] * a^3   ====> 

   M_J = (4* Pi^2*(1.88*10^9)^3)/(6.67 * 10^(-11)*(1.44 * 10^6)^2) in
kg 

   M_J = 1.897 * 10^(27) kg. 


4) Questions about the Sun somehow decreases to half of its current
mass:

a) There would be no effect on your weight. In fact, this does not
depend 
   on the Sun's mass, but on the Earth's mass. 

b) If the Sun became half as massive as it is now, the gravitational 
   force between the Sun and the Earth would be halved. In fact: 

   F = G * (M_s/2 * M_e)/d^2   ====>   F = 1/2 * [G * (M_s * M_e)/d^2]


5) Question about kepler's second law explanation. 
   Answer: 
   In its motion around the Sun, each planet must conserve its angular

   momentum, and the angular momentum L of an object of mass m and 
   velocity r at a distance r from the application point is given by 
   (in vectorial form) L=r*m*v. 
   Considering that the mass remains constant (Newtonian case), then 
   the product of the planet velocity and its distance from the Sun 
   has to be conserved. Therefore at shorter distances from the Sun, 
   each planet will be faster, and this is the meaning of Kepler's 
   second law. 


6) Question about the Earth's mass increased. 
   The two equations to consider are: 

   W = m * g        weight of an object of mass m and 
   F = G * M_e * m / r^2   gravitational force between mass m and
Earth. 

   But the gravitational force on you is just your weight. Therefore 
   if the Earth's mass is increased, also the gravitation force is, 
   and you would weigh more. But your mass would not change.