A rocket leaves Ganymede

[Meistro]

A friend asked me for help on this problem. I don't know much about physics but I did find this place. Any chance you guys could help me?

1: Determine the escape speed of a rocket on the far side of Ganymede, the largest of Jupiter's moons. The radius of Ganymede is 2.64 X 10^6m, and its mass is 1.495 X 10^23 kg. The mass of Jupiter is 1.90 x 10^27 kg, and the distance between Jupiter and Ganymede is 1.071 X 10^9m. Be sure to include the gravitational effect due to jupiter, but you may ignore the motion of Jupiter and Ganymede as they revolve about their center of mass. (U = -GMm/r)

 

[LeonhardEuler]

Alright, if "r" is the distance of the rocket from the center of Ganymede, then the potential energy at position "r" is given by:
$$U=\frac{G*mass of Ganymede*mass of rocket}{r}+\frac{G*mass of Jupiter*mass of rocket}{r+distance between Jupiter and Ganymede}$$
$$=Gm(\frac{M_G}{r}+\frac{M_J}{r+1.071*10^9})$$
(You can just add the distance between Ganymede and Jupiter to "r" because the rocket is on the far side)

So in order to escape, you need to have enough kinetic energy to overcome the potential difference. In our case the potential difference is the difference between U at r=infinity and U at r=radius of Ganymede. So take the difference of the two potentials and set them equal to [itex]\frac{1}{2}mv^2[/tex], and solve for v. The mass of the rocket will cancel from the equation.

 

[kyle8921]

Thank you for reaching out for help with this problem. I am happy to assist you in finding the escape speed of a rocket leaving Ganymede. To solve this problem, we can use the formula for gravitational potential energy (U = -GMm/r) and the conservation of energy principle.

First, we need to calculate the total gravitational potential energy of the rocket on the surface of Ganymede, taking into account the gravitational effect of both Ganymede and Jupiter. This can be done by plugging in the given values for the masses of Ganymede and Jupiter, the distance between them, and the radius of Ganymede into the formula. This will give us the total potential energy at the surface of Ganymede.

Next, we can use the conservation of energy principle to find the escape speed. This principle states that the total energy (potential energy + kinetic energy) of the rocket at the surface of Ganymede must be equal to the total energy of the rocket at the point of escape. We can set the potential energy at the surface of Ganymede equal to the kinetic energy at the point of escape and solve for the escape speed.

I hope this explanation helps and good luck with solving the problem. If you need further assistance, please do not hesitate to ask.