A concept for protecting the earth from asteroids that may threaten it is to hav

Question

A concept for protecting the earth from asteroids that may threaten it is to have a satellite in geosyn- chronous orbit that can fire warheads towards the asteroid to either destroy or divert it. The mass of the warhead is 50 kg, and it is fired by a railgun that can be treated as effecting an instantaneous change in the warhead's velocity. As it approaches its target, the warhead has terminal homing thrusters that can guarantee hitting its target if the trajectory passes within 5 km from the target. The calculation of the trajectory to hit this point from an initial position is an ODE boundary value problem. The equations of motion of warhead can be determined using Newton's law of gravitation and Newton's second law of motion on a particle (assume that this warhead moves within a plane). The force between two bodies due to gravitation is given by m1m2 F=G where G is the gravitational constant (6.674 × 10-11 Nm2/kg, mi is the mass of the first body, m2 is the mass on the second body, and r is the distances between the bodies. For your coordinate system for the problem, place the earth at the origin (0,0) and the launch satellite along the - aris at the radius of the geostationary orbit, which is 42,164km. You may assume that the Earth is stationary during the warhead's flight (a) Determine the equations of motion for the warhead at a position (x, y). Ignore the effect of the force on the earth

Explanation / Answer

given the earth is at origin
so when the satellite is at position (x,y)
distance of the satellite from earth, r = sqroot(x^2 + y^2)
Let mass of the asteroid be m, and its coordinates be (X,Y)
then its distance from the satellite = (x - x, Y - y)
D = sqroot((X - x)^2 + (Y - y)^2)

so in vecttor form
Gravitational force due to earth on the warhead = GM*50(-xi - yj)/r^3 [ where M is mass of earth, i and j are unit vectors in x and y directions]
Gravitational force due to asteroid on warhead = Gm*50((X - x)i + (Y - y)j)/D^3

hence the equation of motion of the warhead becomes
from newtons second law
d^2x i / dt^2 + d^2y j / dt^2 = GM*50(-xi - yj)/r^3 + Gm*50((X - x)i + (Y - y)j)/D^3
d^2x i / dt^2 + d^2y j / dt^2 = GM*50(-xi - yj)/(x^2 + y^2)^3/2 + Gm*50((X - x)i + (Y - y)j)/((X - x)^2 + (Y - y)^2)^3/2

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