Question

Calculate the total energy (in J) needed to launch a satellite of mass 2000 kg from Jupiter surface, that is put in orbit at the same orbital radius as lo (you can assume lo is not close to the satellite, i.e. neglect the gravitational attraction between the satellite and lo).

Answer 1

The total energy needed to launch a satellite from Jupiter's surface and put it in orbit at the same orbital radius as Io can be calculated by considering the potential energy and kinetic energy. The potential energy is determined by the gravitational interaction between Jupiter and the satellite, while the kinetic energy is determined by the satellite's orbital velocity. By calculating these energies and summing them, we can find the total energy required.

Step-by-step explanation:

To calculate the total energy needed to launch a satellite from Jupiter's surface and put it in orbit at the same orbital radius as Io, we need to consider both the gravitational potential energy and the kinetic energy.

The potential energy is given by the equation PE = -(GMm)/r, where G is the gravitational constant (6.67 x 10^-11 Nm^2/kg^2), M is the mass of Jupiter (1.9 x 10^27 kg), m is the mass of the satellite (2000 kg), and r is the distance from the satellite to Jupiter's center.

Since the satellite is put in orbit at the same orbital radius as Io, we can use the formula for the orbital velocity of a satellite, v = sqrt(GM/r), to find the kinetic energy. The total energy is the sum of the potential and kinetic energies.

Using the given information, we can calculate the total energy needed to launch the satellite:

1. 
   
3. Find the distance from Jupiter's surface to its center by subtracting the radius of Jupiter (69,911 km) from the radius of Io's orbit (421,700 km): d = 421,700 km - 69,911 km = 351,789 km = 3.51789 x 10^8 m.
4. 
   
6. Calculate the potential energy: PE = -(6.67 x 10^-11 Nm^2/kg^2)(1.9 x 10^27 kg)(2000 kg)/(3.51789 x 10^8 m) = -2.84 x 10^13 J.
7. 
   
9. Calculate the kinetic energy using the orbital velocity formula: v = sqrt((6.67 x 10^-11 Nm^2/kg^2)(1.9 x 10^27 kg)/(3.51789 x 10^8 m)) = 4.51 x 10^3 m/s. KE = (1/2)(2000 kg)(4.51 x 10^3 m/s)^2 = 2.03 x 10^10 J.
10. 
    
12. Find the total energy: Total energy = PE + KE = -2.84 x 10^13 J + 2.03 x 10^10 J = -2.83797 x 10^13 J.

Therefore, the total energy needed to launch the satellite from Jupiter and put it in orbit at the same orbital radius as Io is approximately -2.83797 x 10^13 J.

Answer 2

Therefore, the total energy needed to launch the satellite from Jupiter's surface to an orbit at the same orbital radius as Io is approximately
`\(2.845 * 10^(12) \, \text{J}\)` in scientific notation.

To calculate the total energy needed to launch a satellite from Jupiter's surface to an orbit at the same orbital radius as Io
`(\(421,700 \, \text{km}\)),` we'll compute the potential energy difference between the surface and the orbit.

The potential energy difference between the surface and the orbit is the work done against gravity:

`\[ PE = G (mM)/(r) \]`

Where:

-
`\( PE \)` is the potential energy.

-
`\( G \)` is the gravitational constant
`(\(6.674 * 10^(-11) \, \text{m}^3/\text{kg s}^2\)).`

-
`\( M \) is the mass of Jupiter (\(1.898 * 10^(27) \, \text{kg}\)).`

-
`\( r \)` is the radius of the orbit.

First, let's convert the orbital radius of Io from kilometers to meters:

`\[ r = 421,700 * 10^3 \, \text{m} = 4.217 * 10^8 \, \text{m} \]`

Now, we'll calculate the potential energy difference:

`\[ PE = G (mM)/(r) \]`

`\[ PE = 6.674 * 10^(-11) \, \text{m}^3/\text{kg s}^2 * \frac{2000 \, \text{kg} * 1.898 * 10^(27) \, \text{kg}}{4.217 * 10^8 \, \text{m}} \]`

Let's solve this:

`\[ PE = 2.845 * 10^(12) \, \text{J} \]`

Question: