Please help me out with these 3 questions or as much as possible!! Thanks

Question

1. does a planet travel faster when it is nearer the sun  or farther from the sun?
2. what is the gravitational attraction between a dating couple sitting 0.152 m apart if the boy weighs 715 N and the girl weighs 465 N?
3. the moon has a mass of 7.35 x 10²² kg. the earths mass is 5.97 x 10²⁴ kg. the distance between the two is 3.84 x 10⁸ m. How far From the earth on a line connecting the earth and the moon would an object have to be for the object to experience equal forces from the earth and the moon? 

 

Thanks in advance for your help!! :)

Answer 1

1. A planet travels faster when it's nearer the sun.

 

2. If P_b=715 N  and  Pg=465 N are the boy's and the girl's weights (correspondingly), then their masses are:

 

m_b=P_b/g  ;  m_g=P_g/g    (g=9.8 N/kg - free fall acceleration)

 

And the gravitational attraction between them is found from Newton's formula:

 

F=G*m_b*m_g/R²     (G=6.67*10^-11 N*m²/kg² - the universal gravitational constant; R=0.152 m - distance between the boy's and the girl's centers of mass)

 

F=G*P_b*P_g/(g*R)² = 9,99*10^-6 N = 10.0 μN.

 

3. The forces acting on a mass m from the Earth (mass M_E) and the moon (mass M_M) are:

 

F_E=G*m*M_E/R_E²    and   F_M=G*m*M_M/R_M² , 

where R_E and R_M are distances from the mass m to the Earth and the moon.

 

From the given conditions: F_E=F_M ;  R_E + R_M = R =3.84*10⁸ m - the distance between the Earth and the moon.

Then: 

 

M_E/R_E²=M_M/R_M²    -->  (R-R_E)²/R_E²=M_M/M_E     -->   R/R_E-1=(M_M/M_E)^1/2    -->  

 

-->     R_E=R/( 1+(M_M/M_E)^1/2 ) = 3.46*10⁸ m.