Calculating v in 0.5mv^2: Right or Wrong?

[UrbanXrisis]

The question is http://home.earthlink.net/~urban-xrisis/clip_image002.jpg

$$0.5mv^2=\frac{GmM_{moon}}{R_{moon}}+\frac{GmM_{jupiter}}{R_{jupiter}+x}$$
x=distance between Jupiter and the moon
$$v=\sqrt{\frac{2GM_{moon}}{R_{moon}}+\frac{2GM_{jupiter}}{R_{jupiter}+x}}$$

$$v=\sqrt{\frac{2*6.673x10^{-11}*1.495E23kg}{2.64E6m}+\frac{2*6.673x10^{-11}*1.9E27kg}{6.99E7+1.071E9}}$$

$$v=17.657km/s$$

when I subbed in the numbers, I get about 17km/s. The books gives 15km/s, am I doing this right?

 

[whozum]

I think you should consider the planets as point masses, particularly Jupiter on the right side of your first line. I would think that the distance 'x' is from center to center.

 

[UrbanXrisis]

that would make the velocity bigger, not smaller, I already tired that but got +18km/s

 

[whozum]

Take into account that the rocket is on the opposite side of the moon from jupiter. so the distance between Jupiter and the rocket would be the distance between them plus a radius of the moon

 

[Gokul43201]

that would make the velocity bigger, not smaller, I already tired that but got +18km/s

Nevertheless, this is the correct answer (~18 km/sec).

You can neglect the radius of Ganymede (it's 3 orders of magnitude smaller) when calculating the distance from Jupe's center.

 

[UrbanXrisis]

Take into account that the rocket is on the opposite side of the moon from jupiter. so the distance between Jupiter and the rocket would be the distance between them plus a radius of the moon

I already tried that too.. the radius of the moon is too small to decrease the escape velocity by 2km/s

Nevertheless, this is the correct answer (~18 km/sec).

You can neglect the radius of Ganymede (it's 3 orders of magnitude smaller) when calculating the distance from Jupe's center.

my book tells me an answer of 15.6 km/s, unless you're saying that it was a missprint?

 

[UrbanXrisis]

$$v=\sqrt{\frac{2GM_{moon}}{R_{moon}}+\frac{2GM_{jupiter}}{R_{jupiter}}}$$

$$v=\sqrt{\frac{2*6.673x10^{-11}*1.495E23kg}{2.64E6m}+\frac{2*6.673x10^{-11}*1.9E27kg}{1.071E9}}$$

$$v=18136m/s$$

 

[Gokul43201]

my book tells me an answer of 15.6 km/s, unless you're saying that it was a missprint?

Either that, or perhaps, one of the numbers provided in the problem is a misprint.

 

[UrbanXrisis]

so you would agree that the escape velocity is indeed 18.136km/s?

 

[Gokul43201]

No, I just redid the calculation and got 15.6 km/sec. Recheck your calculation - step by step. Post numbers here if necessary.

 

[VietDao29]

Distance = Radius of Jupiter + Distance between 2 planets + 2* Radius of Moon.
Viet Dao,

 

[Gokul43201]

Distance = Radius of Jupiter + Distance between 2 planets + 2* Radius of Moon.
Viet Dao,

I don't think so. The specified distance between Jupe and Gany is the distance between their centers.

 

[UrbanXrisis]

$$v=\sqrt{\frac{2GM_{moon}}{R_{moon}}+\frac{2GM_{jupiter}}{R_{jupiter}}}$$

$$v=\sqrt{\frac{2*6.673x10^{-11}*1.495E23kg}{2.64E6m}+\frac{2*6.673x10^{-11}*1.9E27kg}{1.071E9}}$$

what this the right path? I've been punching numbers into my calc for a while now, I don't see how you got 15.6 km/s

 

[whozum]

You're punching numbers wrong, if you have maple, you can do the following:

>v=sqrt((2*G*M)/R+(2*G*N/P));

> subs(G=6.67*10^(-11),M=1.495*10^23,N=1.9*10^27,R=2.64*10^6,P=1.071*10^9,%);

 

[UrbanXrisis]

$$v=\sqrt{\frac{2GM_{moon}}{R_{moon}}+\frac{2GM_{jupiter}}{R_{jupiter}}}$$

$$v=\sqrt{\frac{2*6.673x10^{-11}*1.495E23kg}{2.64E6m}+\frac{2*6.673x10^{-11}*1.9E27kg}{1.071E9}}$$

$$v=\sqrt{\frac{2*6.673x10^{-11}*1.495E23kg}{2.64E6m}}+\sqrt{\frac{2*6.673x10^{-11}*1.9E27kg}{1.071E9}}$$

$$v=\sqrt{\frac{1.995227*10^{13}}{2.64x10^6}}+\sqrt{\frac{2.5357*10^{17}}{1.071x10^9}}$$

$$v=2.749x10^3 + 1.5387x10^4 = 1.8156x10^4$$

still the same answers

 

[VietDao29]

$$\sqrt{a + b} = \sqrt{a} + \sprt{b}$$
Are you sure?
Viet Dao,

 

[UrbanXrisis]

I see my mistake... thanks

that took me way too long

 

[whozum]

You can't split up the square root.

 

[UrbanXrisis]

yeah... what a stupid mistake... arggg

 

[whozum]

dont worry, i once embarassed myself in a class by telling the teacher to split up 1/(x^2+x) into 1/x^2 + 1/x.

Atleast your mistake was infront of 3 and not 40.