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If solve can't find an explicit solution, you can use find_root or the sympy solver:

var ('N,m,sec,a,t,c')

T = (1.27 * 10^6)
Fg = ((9.81*107000))
a = ((T-Fg)/107000)

v = a*t
P3 = plot(v, (t,0,11))

y = v.integral(t)
P1 = plot(y, (t,0,12))

T1 = 1.27*e^(-(t-11)/2) * 10^6
Fg = 9.81*107000
a1 =(T1-Fg)/107000

v1 = a1.integral(t)
v1 = v1 + 154.299065420560
P4 = plot(v1, (t,11,22))

y1 = v1.integral(t) - 1026.68224299065
P2 = plot (y1, (t,12,22))

P = P1 + P2 

Impact = y1.find_root(21,22)
#print Impact

P5 = P3 + P4
show(P5)

dy1 = diff(y1, t)
#print dy1

print 'solve with maxima solver:', solve(v1,t)
print 'solve with find_root:', v1.find_root(0,20)
import sympy
print 'solve with sympy solver:', sympy.solve(v1,t)