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 involved in the launch.
Let's start by calculating the gravitational potential energy:
(a) Equatorial Orbit:
To calculate the energy required to launch a satellite into an equatorial orbit from the Earth's equator, we need to consider the gravitational potential energy change from the Earth's surface to the orbit.
The gravitational potential energy is given by the formula:
PE = mgh
Where:
m = mass of the satellite = 1465 kg
g = acceleration due to gravity = 9.81 m/s^2
h = height above the Earth's surface = 1375 km = 1,375,000 m
Using the given values, we can calculate the change in gravitational potential energy:
PE = 1465 kg * 9.81 m/s^2 * 1,375,000 m
(b) Polar Orbit:
For a polar orbit, we need to consider the gravitational potential energy change from the Earth's surface to the orbit, similar to the equatorial orbit calculation.
Using the same formula as above and substituting the values, we can calculate the change in gravitational potential energy for the polar orbit.
Next, let's calculate the kinetic energy involved in the launch:
The kinetic energy is given by the formula:
KE = (1/2) * m * v^2
Where:
m = mass of the satellite = 1465 kg
v = velocity of the satellite
To calculate the velocity of the satellite, we need to apply the formula for the orbital velocity:
v = √(GM/R)
Where:
G = gravitational constant = 6.67 x 10^-11 N(m/kg)^2
M = mass of the Earth = 5.97 x 10^24 kg
R = radius of the Earth + height of the orbit
(a) Equatorial Orbit:
For the Equatorial Orbit, we can calculate the velocity by considering the radius of the Earth plus the height of the orbit.
(b) Polar Orbit:
For the Polar Orbit, we need to consider the radius of the Earth plus the height of the orbit from the North Pole.
Once we have calculated the velocity, we can substitute it into the kinetic energy formula to find the kinetic energy.
Finally, to estimate the total energy required for the launch, we can add the change in gravitational potential energy and the kinetic energy for each case.
Please note that these calculations are approximations and do not account for factors like air resistance and the variation of gravitational force with height.