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two intervals const a p3
Position
x
(
t
)
=
{
x
0
+
v
0
(
t
−
t
0
)
+
1
2
a
0
(
t
−
t
0
)
2
,
t
0
≤
t
<
t
1
,
x
1
+
v
1
(
t
−
t
1
)
+
1
2
a
1
(
t
−
t
1
)
2
,
t
1
≤
t
<
t
2
,
(D3)
\tag{D3} x(t) = \left\{ \begin{array}{cc} x_0 + v_0(t - t_0) + \tfrac12 a_0(t - t_0)^2, & t_0 \le t < t_1, \\[0.5em] x_1 + v_1(t - t_1) + \tfrac12 a_1(t - t_1)^2, & t_1 \le t < t_2, \end{array} \right.
x
(
t
)
=
{
x
0
+
v
0
(
t
−
t
0
)
+
2
1
a
0
(
t
−
t
0
)
2
,
x
1
+
v
1
(
t
−
t
1
)
+
2
1
a
1
(
t
−
t
1
)
2
,
t
0
≤
t
<
t
1
,
t
1
≤
t
<
t
2
,
(
D3
)
with
t
n
+
1
=
t
n
+
τ
n
t_{n+1} = t_n + \tau_n
t
n
+
1
=
t
n
+
τ
n
, where
τ
n
\tau_n
τ
n
is
n
n
n
-th time interval.
Initial conditions
x
(
t
0
)
=
x
0
x(t_0) = x_0
x
(
t
0
)
=
x
0
,
x
(
t
1
)
=
x
1
x(t_1) = x_1
x
(
t
1
)
=
x
1
and
x
(
t
1
)
=
x
0
+
v
0
(
t
1
−
t
0
)
+
1
2
a
0
(
t
1
−
t
0
)
2
x(t_1) = x_0 + v_0(t_1 - t_0) + \tfrac12 a_0(t_1 - t_0)^2
x
(
t
1
)
=
x
0
+
v
0
(
t
1
−
t
0
)
+
2
1
a
0
(
t
1
−
t
0
)
2
.