NDSolve will try solving the system as differential-algebraic equations but it didn't get the solution
Another solution is to use Simplify`PWToUnitStep
:
s1 = NDSolve[{equa00 // Simplify`PWToUnitStep, x[0] == 1, x'[0] == 1}, x, {t, 0, 50}]
Changing the last line to:
s1 = NDSolve[{equa00, x[0] == 1, x'[0] == 1}, x, {t, 0, 50}, SolveDelayed -> True]
or
s1 = NDSolve[{equa00, x[0] == 1, x'[0] == 1}, x, {t, 0, 50},
Method -> {"EquationSimplification" -> "Residual"}]
seems help for your problem.
In reponse to updated question on plot slution
To plot your solution, maybe this is what you want?
Remove["Global`*"] // Quiet;
m = 100;
a = D[x[t], {t, 2}];
t1up = 2 x''[t] + 1/2 (490 + 34 x''[t] + 2 (490 + 50 x''[t]));
t1down = 490 + 53 x''[t];
t1 = Piecewise[{{t1up, x'[t] >= 0}, {t1down, x'[t] < 0}}];
equa00 = t1 == m*a;
t0 = 50;
(*s1 = NDSolveValue[{equa00 // Simplify`PWToUnitStep, x[0] == 1,
x'[0] == 1}, x, {t, 0, 50}];*)
s1 = x /.First@NDSolve[{equa00 // Simplify`PWToUnitStep, x[0] == 1,
x'[0] == 1}, x, {t, 0, 50}];
sAll = {x[t] -> s1[t], x'[t] -> s1'[t], x''[t] -> s1''[t]};
t1upvalue = t1up /. sAll;
t1downvalue = t1down /. sAll;
t1value =
Piecewise[{{t1upvalue, s1'[t] >= 0}, {t1downvalue, s1'[t] < 0}}];
Plot[t1value, {t, 0, t0}, PlotRange -> All]