Edit If we want to use v1 and v2,we can use With to enclose WhenEvent

Edit

If we want to use v1 and v2,we can use With to enclose the WhenEvent

When we remove v1,v2, all of them work fine.

Edit If we want to use v1 and v2,we can use With to enclose WhenEvent

Clear["Global`*"];
(*Preliminary definition*)ClearAll[ip, v1, v2];
ip[t_] := 
  Piecewise[{{Sin[2 π t/(2 0.1)], t < 0.1}, {0, 0.1 <= t}}];
(*The differential equations*)
eqn = {986. y[t] + 6. Derivative[1][y][t] + y''[t] == 
    986 ip[t] + 6. ip'[t], z''[t] == -9.81`, 
   With[{v1 = 0.9 (0.95 y'[t] + 0.15 z'[t]), 
     v2 = 0.9 (1.5 y'[t] - 0.4 z'[t])}, 
    WhenEvent[{z[t] - y[t] <= -0.125, 
      z[t] - y[t] >= 0.125}, {Derivative[1][y][t] -> v1, 
      Derivative[1][z][t] -> v2}]], y[0] == 0, 
   Derivative[1][y][0] == 0, z[0] == 0, Derivative[1][z][0] == 0};

(*Sove the equation*)
sol = First@NDSolve[eqn, {y, z}, {t, 0, 0.04}];

(*Plot the result*)
Plot[Evaluate[z[t] - y[t] /. sol], {t, 0, 0.04}, 
 Epilog -> {Pink, InfiniteLine[{0, -0.125}, {1, 0}]}]

Original

When we remove v1,v2, all of them work fine.