647.sagews

pretty_print_default() 

var('x_b1,y_b1,z_b1,v_x1,v_y1,v_z1,t')

eq = (x_b1 + v_x1*t)*v_x1 + (y_b1 + v_y1*t)*v_y1 + (z_b1 + v_z1*t)*v_z1 == 0

a = solve(eq,t)
print "min time:"

a[0]

min time:

\newcommand{\Bold}[1]{\mathbf{#1}}t = -\frac{{(v_{x_{1}} x_{b_{1}} + v_{y_{1}} y_{b_{1}} + v_{z_{1}} z_{b_{1}})}}{{(v_{x_{1}}^{2} + v_{y_{1}}^{2} + v_{z_{1}}^{2})}}

pretty_print_default() 
var('dx,dy,dz,dvx,dvy,dvz')

var('t')
var('h_nmac')

eq_c = (dx-dvx*t)^2+(dy-dvy*t)^2==h_nmac

a = solve(eq_c,t)

print "entry and exit times:"

print a[0]
print a[1]

entry and exit times: t == (dvx*dx + dvy*dy - sqrt(-dvx^2*dy^2 + 2*dvx*dvy*dx*dy - dvy^2*dx^2 + (dvx^2 + dvy^2)*h_nmac))/(dvx^2 + dvy^2) t == (dvx*dx + dvy*dy + sqrt(-dvx^2*dy^2 + 2*dvx*dvy*dx*dy - dvy^2*dx^2 + (dvx^2 + dvy^2)*h_nmac))/(dvx^2 + dvy^2)

var('t')

r_nmac = 1500

tr  = 100;
i_t = 45*pi/180;
i_v = -200;

xo = 0
yo = 0
vxo = 200
vyo = 0

xi = (vxo-i_v*cos(i_t))*tr
yi =     -i_v*sin(i_t) *tr
vxi = cos(i_t)*i_v
vyi = sin(i_t)*0

eq_s = (-(xo+vxo*t)+(xi+vxi*t))^2+(-(yo+vyo*t)+(yi+vyi*t))^2==r_nmac^2

a = solve(eq_s,t)

print "entry and exit times:"

print eq_s

entry and exit times: 10000*(sqrt(2)*t + 2*t - 100*sqrt(2) - 200)^2 + 200000000 == 2250000