A planet orbits a star in an elliptical orbit At a particula

A planet orbits a star in an elliptical orbit. At a particular instant the momentum of the planet is <2.8 10^29, 1.6 10^29, 0> kg · m/s, and the force on the planet by the star is <2.7 10^22, 1.0 10^23, 0> N. Find F and F . (Express your answers in vector form.)

Solution

The momentum of the particle :

     P = <2.8 10^29, 1.6 10^29, 0> kg · m/s

Magnitude of the momentum:

   |P| = sqrt[(2.8 10^29)^2 +(1.6 10^29)^2+0 ] = 3.2249e+29 kg.m/s

unit vector of momentum is,

   P/|P| = { <2.8 10^29, 1.6 10^29, 0>} / {3.2249e+29 }

           = < -0.868, -0.496,0> kg.m/s

Parallel force:

    F_|| = (F.P^)(P^)

        = [(<2.7 10^22, 1.0 10^23, 0>).( { < -0.868, -0.496,0> kg.m/s})]{ < -0.868, -0.496,0> kg.m/s}

        = <-2.03 10^22, -2.46 10^22, 0> N

magnitude: F|| = sqrt[(-2.03 10^22)^2+( -2.46 10^22)^2 + 0>] = 3.16x10²² N

Perpendicular force:

     F = F - F_|| = {<2.7 10^22, 1.0 10^23, 0> - <-2.03 10^22, -2.46 10^22, 0>}

                         = <-0.67 10^22, -7.54 10^22,0> N

Magnitude: F = 7.57x10²² N