A satellite is given a boost of 540 MJ of energy to move it from its initial orbit at an altitude of 150 km to a higher altitude orbit. If the satellite has a mass of 1.16 ✕ 10^3 kg, what is the new altitude it reaches? Take the mass of the Earth to be 5.97 x 10^24 kg and its radius to be 6.371 x 10^6 m.
The energy required to move the satellite from one orbit to another can be expressed as the difference in the gravitational potential energy (GPE) between the two orbits. The GPE is given by:
GPE = -GMm / r,
where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance from the center of the Earth.
Initially, the satellite is at an altitude of 150 km, so its initial distance from the center of the Earth is the sum of the Earth's radius and the altitude:
r1 = 6.371 x 10^6 m + 150 x 10^3 m = 6.521 x 10^6 m.
Let the new altitude be h, then the distance from the center of the Earth in the new orbit is:
r2 = 6.371 x 10^6 m + h.
The energy difference required for the satellite to move to the new orbit is given as 540 MJ or 540 x 10^6 J. Therefore, we can write:
ΔGPE = GPE2 - GPE1 = 540 x 10^6 J.
Now we can substitute the expressions for GPE and solve for h:
- GMm / r2 + GMm / r1 = 540 x 10^6 J.
We can cancel out the satellite mass m and gravitational constant G, and rearrange the equation to solve for r2:
r2 = 1 / [(1 / r1) - (540 x 10^6 J) / (5.97 x 10^24 kg * 6.674 x 10^(-11) m^3/kg/s^2) ],
where we have substituted the values for Earth's mass M and gravitational constant G.
Now we can calculate the result:
r2 = 1 / [(1 / (6.521 x 10^6 m)) - (540 x 10^6 J) / (5.97 x 10^24 kg * 6.674 x 10^(-11) m^3/kg/s^2)]
=> r2 ≈ 7.00 x 10^6 m.
Finally, we can find the new altitude h by subtracting the Earth's radius:
h = r2 - 6.371 x 10^6 m = 7.00 x 10^6 m - 6.371 x 10^6 m = 0.629 x 10^6 m ≈ 629 km.
So the new altitude reached by the satellite after the energy boost is approximately 629 km.
To determine the new altitude of the satellite, we first need to calculate the initial total energy (E1) and the final total energy (E2) of the satellite.
The total energy of an object in orbit is the sum of its kinetic energy and its gravitational potential energy.
The initial kinetic energy (KE1) of the satellite can be calculated using the formula:
KE1 = (1/2)mv^2
where m is the mass of the satellite and v is its velocity in the initial orbit.
The velocity in the initial orbit can be found using the formula:
v = √(GM/r)
where G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), M is the mass of the Earth, and r is the radius of the initial orbit.
The initial gravitational potential energy (PE1) of the satellite can be calculated using the formula:
PE1 = -GMm/r
where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the radius of the initial orbit.
The initial total energy (E1) is the sum of the initial kinetic energy (KE1) and the initial gravitational potential energy (PE1).
E1 = KE1 + PE1
Now, let's calculate the initial total energy (E1):
1. Calculate the velocity in the initial orbit (v):
- G = 6.67430 x 10^-11 m^3 kg^-1 s^-2
- M = 5.97 x 10^24 kg
- r = radius of the initial orbit = radius of the Earth + altitude of the initial orbit
Substitute these values into the formula to calculate v.
2. Calculate the initial kinetic energy (KE1):
- Use the mass of the satellite (m) and the velocity (v) calculated in the previous step.
- Substitute these values into the formula to calculate KE1.
3. Calculate the initial gravitational potential energy (PE1):
- Use the values of G, M, m, and r.
- Substitute these values into the formula to calculate PE1.
4. Calculate the initial total energy (E1):
- Sum the values of KE1 and PE1.
Now that we have the initial total energy (E1), we can calculate the final total energy (E2) using the equation:
E2 = E1 + ΔE
where ΔE is the change in energy given to the satellite.
In this case, the change in energy (ΔE) is given as 540 MJ, which needs to be converted to joules before adding it to E1.
1 MJ = 1 x 10^6 J
So, ΔE = 540 MJ x (1 x 10^6 J / 1 MJ)
Finally, we can calculate the new altitude:
1. Rearrange the formula for gravitational potential energy to solve for r:
PE2 = -GMm/r
2. Rearrange the formula for total energy to solve for r:
E2 = KE2 + PE2
Substituting the values of KE2 and PE2, we get:
E2 = (1/2)m(v2^2) - GMm/r
where v2 is the velocity in the new orbit.
Rearranging, we get:
GMm/r = (1/2)m(v2^2) - E2
3. Rearrange the formula for velocity to solve for v2:
v2 = √((2GM)/r + (2E2/m))
4. Substitute the values of G, M, E2, and m to calculate v2.
5. Calculate the new altitude using the formula:
r = (r' - R) + R
where r' is the new orbital radius, and R is the radius of the Earth.
Remember to convert the final answer back to meters from the new orbital radius.
By following these steps, you should be able to calculate the new altitude the satellite reaches.