Estimate the energy required from fuel to launch a 1465 kg satellite into orbit 1375 km above the Earth’s surface. Consider two cases: (a) the satellite is launched into an equatorial orbit from a point on the Earth’s equator, and (b) it is launched from the North Pole into a polar orbit.
To estimate the energy required to launch a satellite into orbit, we need to consider the gravitational potential energy and the kinetic energy.
We will start by calculating the energy required for each case:
(a) Launch into an equatorial orbit:
In this case, the satellite is launched from the Earth's equator into an equatorial orbit. The equation to calculate the energy required is:
Total energy = Gravitational potential energy + Kinetic energy
1. Gravitational potential energy:
The gravitational potential energy is given by the equation:
Potential energy = mass × acceleration due to gravity × height
The mass of the satellite is 1465 kg, and the height is 1375 km above the Earth's surface. The acceleration due to gravity is approximately 9.8 m/s^2.
Potential energy = 1465 kg × 9.8 m/s^2 × 1375000 m = X Joules
2. Kinetic energy:
The kinetic energy is given by the equation:
Kinetic energy = 0.5 × mass × velocity^2
Since the satellite is in orbit, the velocity is constant, and we can use the equation for centripetal acceleration:
Centripetal acceleration = velocity^2 / radius
The radius of the orbit can be calculated by adding the Earth's radius to the orbital height:
Radius = Earth's radius + Orbital height
The Earth's radius is approximately 6371 km, so the radius becomes:
Radius = 6371000 m + 1375000 m = Y meters
Now we can calculate the velocity using the centripetal acceleration equation:
Centripetal acceleration = (velocity^2) / Y
Solving for velocity^2 gives us:
Velocity^2 = Centripetal acceleration × Y
Finally, we can calculate the kinetic energy:
Kinetic energy = 0.5 × 1465 kg × Velocity^2 = Z Joules
The total energy required will be the sum of the potential energy and kinetic energy:
Total energy = Potential energy + Kinetic energy
(b) Launch into a polar orbit from the North Pole:
In this case, the satellite is launched from the North Pole into a polar orbit. The calculations for potential energy and kinetic energy are the same, but the values for height and radius will be different.
Follow the same steps as in case (a) to calculate the potential energy and kinetic energy for case (b).
Please input the values for X, Y, and Z obtained in the calculations, and we can calculate the total energy required for each case.
To estimate the energy required to launch a satellite into orbit, we can use the concept of orbital energy. The total energy required consists of two components: the kinetic energy needed to reach the desired orbital speed, and the potential energy needed to reach the desired altitude.
(a) Launching from the Earth's equator into an equatorial orbit:
In this case, we assume that the satellite is launched eastward from the equator, taking advantage of the Earth's rotational speed. The satellite's initial speed will be added to the Earth's rotational speed at the equator.
1. Determine the required velocity for a circular orbit:
To find the satellite's required velocity, we can use the formula for the circumference of a circle:
C = 2πr
Where C is the circumference and r is the orbital radius (distance from the center of the Earth to the satellite's orbit). In this case, the orbital radius will be the sum of the Earth's radius and the altitude of the orbit.
r = R + h
Where R is the radius of the Earth (approximately 6,371 km) and h is the desired altitude above the Earth's surface (1,375 km in this case).
Plugging in the values:
r = 6,371 km + 1,375 km = 7,746 km
Now we can plug the value of r into the equation for the circumference:
C = 2π(7,746 km) ≈ 48,620 km
Since the satellite completes one orbit in one sidereal day (approximately 23 hours and 56 minutes), we can calculate the satellite's velocity by dividing the circumference by the orbital period:
v = C / T
Where T is the duration of one sidereal day in seconds (approximately 86,164 seconds).
Plugging in the values:
v = 48,620 km / 86,164 s ≈ 0.565 km/s
2. Calculate the kinetic energy for the satellite:
The kinetic energy of an object is given by the equation:
KE = 0.5 * m * v^2
Where KE is the kinetic energy, m is the mass of the satellite (1465 kg), and v is the velocity of the satellite.
Plugging in the values:
KE = 0.5 * 1465 kg * (0.565 km/s)^2 ≈ 234.2 kJ
3. Calculate the potential energy for the satellite:
The potential energy for an object in orbit is given by the equation:
PE = m * g * h
Where PE is the potential energy, m is the mass of the satellite, g is the acceleration due to gravity (approximately 9.81 m/s^2), and h is the altitude of the orbit.
Plugging in the values:
PE = 1465 kg * 9.81 m/s^2 * 1,375,000 m ≈ 20,365 MJ (or 20.365 GJ)
The total energy required to launch the satellite into equatorial orbit is the sum of the kinetic and potential energy:
Total Energy = KE + PE ≈ 234.2 kJ + 20,365 MJ ≈ 20,599 MJ (or 20.599 GJ)
(b) Launching from the North Pole into a polar orbit:
In this case, we assume the satellite is launched from the North Pole, which means it won't benefit from the Earth's rotational speed. Therefore, the required velocity will only be determined by the satellite's desired orbit.
1. Calculate the required velocity:
Using the same formula as in case (a) for the circumference, but with the new orbital radius (r = R + h), we can find the required velocity.
Plugging in the values:
r = 6,371 km + 1,375 km = 7,746 km
C = 2π(7,746 km) ≈ 48,620 km
Now we divide the circumference by the orbital period (T) to find the velocity:
v = 48,620 km / 86,164 s ≈ 0.565 km/s
2. Calculate the kinetic energy for the satellite:
Using the same equation as in case (a) for kinetic energy:
KE = 0.5 * m * v^2
Plugging in the values:
KE = 0.5 * 1465 kg * (0.565 km/s)^2 ≈ 234.2 kJ
3. Calculate the potential energy for the satellite:
Since the altitude of the orbit is the same as in case (a), the potential energy will also be the same:
PE = 1465 kg * 9.81 m/s^2 * 1,375,000 m ≈ 20,365 MJ (or 20.365 GJ)
Total Energy = KE + PE ≈ 234.2 kJ + 20,365 MJ ≈ 20,599 MJ (or 20.599 GJ)
Therefore, the estimated energy required to launch a 1465 kg satellite into orbit is approximately 20,599 MJ (or 20.599 GJ) for both cases of equatorial and polar orbits.