The figure below shows three isolated planets, A, B and C on a grid where the sp
ID: 2095434 • Letter: T
Question
The figure below shows three isolated planets, A, B and C on a grid where the spacing is a. The mass of the planets is M, M and 2M respectively. There is a small spaceship of mass m located at point P. Assume that the planets are fixed in position and the spaceship is initially at rest. Assume that the only force acting on the spaceship is the gravitational force due to the three planets. (a) What are the vector components of the net gravitational force acting on the spaceship at P? (b) The spaceship fires its rocket briefly so that it is moving in the +z direction and then it turns off its engine so it coasts ever after. What minimum velocity does the ship need to achieve to escape from the three-planet system? (c) How much energy would it take to separate the planets so that each is an infinite distance from the other two?Explanation / Answer
assume that planet c is origin so coordinates of A (2,3,0) B(-1,3,0) c(0,0,0) P(2,0,0) center of mass of three planet X coordinate = 2M-M/4M = 1/4 Y coordinate = 3M+3M/4M = 1.5 so equivalent mass system = (.25,1.5,0) of 4M mass A) f = [G(m) 4M /5.31] [ (1.75i-1.5j)/2.31 ] answer B) potential energy due to system = G*4M / 5.31 so we have to give min. kE which is equal to G*4M / 5.31 so 0.5 m*v^2 = G*4M / 5.31 so v= 1.23 sqrt(GM/m) c) = GM/(bc)^2 +GM/(ac)^2 +GM/(ab)^2