Mass of Planet X = 4.96*10^24 kg Radius of Planet X = 7.69*10^3 km Planet X also
ID: 1570746 • Letter: M
Question
Mass of Planet X = 4.96*10^24 kg
Radius of Planet X = 7.69*10^3 km
Planet X also has a strange electrical characteristic...it has an electrical charge density of 1.00 fC / m3, uniform throughout its volume! This leads to a lot of lightning storms. And it also means that the Xians have devised a neat way to launch their spaceships...they can turn a nifty (and somewhat magical) knob to control the electrical charge of their ships. (Conveniently, the atmosphere of X is very thin so that air resistance can be ignored!)
a. If an X ship (including crew and payload) has a mass of 10,000 kg, how much net charge should the crew give the ship if an initial launch acceleration at the planet’s surface of a = gx is desired (where gx is the acceleration due to gravity at the surface of X)?
b. If they want to maintain this acceleration throughout the launch, what should the charge of the ship be when the ship is Rx above the surface? What is the electric field due to X at the surface?
c. Assuming the electric potential is zero at infinity, what is the electric potential at the surface of X?
d. And what are the field and potential at Rx above the surface? Now the ship is returning to X. It has a downward velocity of 10.0 km/s at a height Rx above the surface. (Assume that the gravitational potential is also zero at infinity.)
e. If the crew wants the ship to come gently, perfectly to rest as it lands on the surface, what charge should they give the ship at the altitude Rx above the surface? As always, show all of your calculations, provide useful sketches, and explain your approach and results. Please show your work! Thanks!
Explanation / Answer
mass of planet Mx = 4.96 e+24 kg
radius Rx = 7.69e+3 km
charge density 1.0fC /m3
volume of the planet V = 4piRx3/3 = 1.905 e+21 m3
Total charge on the planet Qx = 1.905e+21 *1.0e-15 = 1.905e+6 C
a) Mass of the spacc ship m = 10000 kg
gravitational force = mgx
required acceleration a = gx
let q be the charge on the space ship, then
culomb force Fc = kQxq/R2
equation of motion
ma = kQxq/R2 - mgx
q = 2mgx R2 /kQx
The gravitational force is
mgx = GMxm/R2
gx = GMx/R2
q = 2mGMx /kQx = 2*10000*6.67e-11 *4.96e+24 /9.0e+9*1.90e+6
= 386.9 C
when the ship is Rx above the surface ( it is differet from R we used ofr radius of the planet)
the eq. of motion still remains same
q = 2mgx Reff2 /kQx where Reff = R+ Rx
and mgx = GMx m /Reff2
gx Reff2 = GMx
q = 2mGM/kQx , it is independent of the distance Rx and the charge requried remains same.
c) electric potential at the surface of the planet
V = kQ/R = 9.9e+9 *1.9e+6/ 7.69e+6 = 2.44 e+9V