Could anyone break this problem down for me. A satellite starts in a circular or
ID: 2116257 • Letter: C
Question
Could anyone break this problem down for me.
A satellite starts in a circular orbit of radius r ( = distance from the center of the Earth). Its speed is v1. The gravitational force on the satellite is central. Fr = - GMm/r2 where M is the mass of the Earth; the potential energy is -GMm/r. Suddenly, a burst of the rocket engine increases the speed of the satellite from v1 to v1+0.3 v1. Thereafter the satellite moves on an elliptical orbit, as shown. Calculate the apogee ra (= greatest distance from the Earth). Express the answer in terms of the initial radius r. (Show your work!)Explanation / Answer
mv1 ^ 2 / r = F = GMm/r^2
v1= sqrt (GM / r)
let apogee distance be x
velocity * distance from earth = constant {this is a formula from chapter gravitation}
(1.3)v1 * r = v * x
by conservation of energy
1/2 m(1.3v1)^2 - GMm/r = 1/2 m v^2 - GMm/x
mutiplyling whole equation with x^2
x^2 [ (1.3 v1)^2 - GM/r ] = (vx)^2 - GMx
x^2 [ (1.3 v1)^2 - GM/r ] = (v1*r)^2 -GMx
x^2 [ (1.3 v1)^2 - GM/r ]= GMr - GMx
x^2 [ (1.3)^2 * GM/r - GM/r ]= GMr - GMx
taking GM common and cancling
x^2 [ (1.3)^2 *1/r - 1/r ]= r - x
solving quadratic in x we get
x= 0.68 r