Finding r of a Jovian-synchronous orbit and escape velocity

[AN630078]

Homework Statement

Hello, I have recently been introduced to the principle of escape velocity, in additional to other gravitational concepts, and finding the orbital period. I was attempting some practise questions I found online but I wondered whether anyone could comment upon my workings and calculations, particularly to suggest any improvements if I am faced with similar problems. I would be very grateful of any help

1.The mass of Jupiter is 1.89 x 10^27 kg. It rotates on its own axis in 9 hours and 55 minutes. At
what distance from Jupiter would a satellite above the equator remain always above the same spot?
2. The radius of Jupiter is 7.1 x 107. Calculate the escape velocity from Jupiter

Relevant Equations

T^2=4π^2r^3/GM
Ui + Ki=Uf+Kf

1. The satellite would be in a jovian-synchronous orbit,
Rearranging the formula for the orbital period in terms of r, since T^2 is proportional to r^3:
T^2=4π^2r^3/GM which becomes r^3=(GM/4π^2) T^2

M(mass of Jupiter)=1.89 x 10^27
G=6.67*10^-11 m^3kg^-1s^-2
T=9 hours and 55 minutes = 3.57*10^4 s

Therefore, r^3= (6.67*10^-11*1.89 x 10^27 /4π^2) *(3.57*10^4)^2
r^3=4.069718155*10^24 m
Taking the cube root of both sides;
r=159657053.9 ~ 160,000,000 m or 1.6*10^8 m

So the satellite would need to remain 1.6*10^8 m above the equator of Jupiter.

2. The escape velocity is the minimum speed required for a free, non-propelled object to escape from the gravitational influence of a massive body, that is, to achieve an infinite distance from it.

Applying the conservation of energy;
Ui + Ki=Uf+Kf
Set both terms on the right to zero, since we want the object to reach infinity, where the potential energy is zero.
For a planet of mass M and radius R, the potential energy of an object of mass m at the planet's surface is: U= - GMm/R.
Therefore, - G Mm/R + K escape =0
1/2mv^2 escape = GMm/R
Cancel m and multiply both sides by 2:
v escape = √ 2GM/R
v escape =√ 2*6.67*10^-11*1.89 x 10^27 /7.1 x 10^7
v escape = 59590.85855 ~60,000 ms^-1

 

[TSny]

Looks good for both parts. But for part 1 you concluded that

r=159657053.9 ~ 160,000,000 m or 1.6*10^8 m

So the satellite would need to remain 1.6*10^8 m above the equator of Jupiter.

Does r represent the distance above the surface of Jupiter or the distance from the center of Jupiter?

 

[AN630078]

Looks good for both parts. But for part 1 you concluded that
Does r represent the distance above the surface of Jupiter or the distance from the center of Jupiter?

Thank you for your reply. I was using r to represent the distance above the surface of Jupiter I believe, would this be wrong?

 

[TSny]

Thank you for your reply. I was using r to represent the distance above the surface of Jupiter I believe, would this be wrong?

r denotes the radius of the circle that represents the orbit of the satellite. It's not really clear to me what the question means by "distance from Jupiter".

 

[AN630078]

r denotes the radius of the circle that represents the orbit of the satellite. It's not really clear to me what the question means by "distance from Jupiter".

I think the "distance from Jupiter" is supposed to be the distance from the equator at the surface of Jupiter to the height of the orbiting satellite. Would be workings be incorrect then?

 

[TSny]

I think the "distance from Jupiter" is supposed to be the distance from the equator at the surface of Jupiter to the height of the orbiting satellite. Would be workings be incorrect then?

I believe that you got the correct answer for r. But r is not the distance between the surface of Jupiter and the satellite.

 

[AN630078]

I believe that you got the correct answer for r. But r is not the distance between the surface of Jupiter and the satellite.

Thank you for your reply. Oh ok, should I try to amend this or is it the wording of the question that is erroneous? Also, would my calculations for the escape velocity be correct?

 

[TSny]

Thank you for your reply. Oh ok, should I try to amend this or is it the wording of the question that is erroneous? Also, would my calculations for the escape velocity be correct?

For me, the problem statement for part 1 is not too clear. The fact that they don't give you the radius of Jupiter until part 2 makes me think that they are just asking for r in part 1, rather than asking for the height above the surface of Jupiter. But, I'm not sure. Since this is just a practice problem, I wouldn't worry about it as long as you understand the meaning of r and how you could use r and the radius of the planet to calculate the height above the surface.

Your work for part 2 looks good to me.