Derive an expression for escape speed of satellite.
KE needed=PE at origin
1/2 m v^2=GMe*m/distance
"escape speed of satellite" would depend on escape from what, I assume you are trying to get it out of orbit into deep space. https://en.wikipedia.org/wiki/Escape_velocity
To derive the expression for the escape speed of a satellite, we need to consider the gravitational force acting on the satellite and apply energy conservation.
Let's assume we have a satellite of mass m orbiting a planet of mass M. The distance between the center of the planet and the satellite is r. The escape speed, denoted by v, is the minimum speed needed for the satellite to escape the planet's gravitational pull and move away to an infinitely far distance.
To derive the expression, we will consider the total mechanical energy of the satellite, which is the sum of its kinetic energy (KE) and potential energy (PE).
1. The kinetic energy of the satellite is given by KE = (1/2)mv^2.
2. The potential energy of the satellite is given by PE = - (GmM)/r, where G is the gravitational constant.
Now, at the satellite's initial position in orbit, the total mechanical energy is the sum of its kinetic and potential energy, which can be written as:
E = KE + PE = (1/2)mv^2 - (GmM)/r
At the escape point, the satellite has just enough energy to overcome the gravitational attraction of the planet. In other words, at escape, the total mechanical energy is zero (E = 0), since the satellite is moving at the escape speed to reach infinitely far from the planet.
Setting E = 0, we can solve for the escape speed (v):
(1/2)mv^2 - (GmM)/r = 0
Simplifying this equation:
(1/2)mv^2 = (GmM)/r
Now, we can solve for v:
v^2 = (2GM)/r
Taking the square root of both sides, we get:
v = sqrt((2GM)/r)
So, the expression for the escape speed of a satellite is given by:
v = sqrt((2GM)/r)
Where:
- v is the escape speed of the satellite,
- G is the gravitational constant (approximately 6.674 × 10^(-11) m^3 kg^(-1) s^(-2)),
- M is the mass of the planet,
- r is the distance between the center of the planet and the satellite.
By plugging in the appropriate values for G, M, and r, you can calculate the escape speed of a satellite for a given planet.