Hi Khalid,

I rewrite your original equation a bit:

In[11]:= eq1 = m1 x''[t] == - (x[t])^((3/2))

We see the right hand term is a restoring force. When x[t] is positive, it applies a negative force to send x in the negative direction. If it's a real spring, then when x[t] goes negative, it should be applying a positive force. But it doesn't. Instead it produces a complex number.

So I changed the physics:

eq2 = x''[t] == -Sign[x[t]] Abs[x[t]]^(3/2)

Now the term on the right hand side works for both positive and negative x[t]. It has the same nonlinear relation on each side of zero, with the proper change of sign. Look up the functions Sign and Abs. Here we plot the restoring force term.

Plot[-Sign[x[t]] Abs[x[t]]^(3/2), {x[t], -1, 1}, 
 AxesLabel -> {"Restoring Force", "x[t]"}]

Note that in revising the notebook, WhenEvent is not needed. It is the physics, or if you like the math description of the physics, which I revised. Not the code.

Kind regards, David

The equation from your notebook is:

EQ1 = m1 x1''[t] + (x1[t])^(3/2) == 0 /. values

You begin with x1'[t] and x1[t] positive, but the force term drives x1'[t] negative and eventually x1[t] goes negative. This is when (x1[t])^(3/2) becomes complex. If this is a projectile, it just hit the ground and you could use "StopIntegration" in WhenEvent.

If it is really a spring which needs a real restoring force when x1 < 0, then the mathematics needs a different formulation. For example:

EQ1 = m1 x1''[t] + Sign[x1[t]] Abs[x1[t]]^(3/2) == 0 /. values

which calculates a real spring force, but also preserves the sign of the force.

Kind regards, David