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kinematics, integrate v to x
Integrate velocity
v
v
v
to obtain position
x
x
x
x
(
t
)
−
x
(
t
0
)
=
∫
t
0
t
[
v
0
+
∫
t
0
τ
2
a
(
τ
1
)
d
τ
1
]
d
τ
2
.
(F1)
\tag{F1} x(t) - x(t_0) = \int_{t_0}^t \left[ v_0 + \int_{t_0}^{\tau_2} a(\tau_1) d\tau_1 \right] d\tau_2.
x
(
t
)
−
x
(
t
0
)
=
∫
t
0
t
[
v
0
+
∫
t
0
τ
2
a
(
τ
1
)
d
τ
1
]
d
τ
2
.
(
F1
)
Initial condition
x
(
t
0
)
=
x
0
x(t_0) = x_0
x
(
t
0
)
=
x
0
, will give
x
(
t
)
=
x
0
+
v
0
(
t
−
t
0
)
+
∫
t
0
t
∫
t
0
τ
2
a
(
τ
1
)
d
τ
1
d
τ
2
.
(F2)
\tag{F2} x(t) = x_0 + v_0 (t - t_0) + \int_{t_0}^t \int_{t_0}^{\tau_2} a(\tau_1) d\tau_1 d\tau_2.
x
(
t
)
=
x
0
+
v
0
(
t
−
t
0
)
+
∫
t
0
t
∫
t
0
τ
2
a
(
τ
1
)
d
τ
1
d
τ
2
.
(
F2
)