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the 3rd kine eqns, p2
From previous equations following can be obtained
x
=
x
0
+
v
0
(
v
−
v
0
a
)
+
1
2
a
(
v
−
v
0
a
)
2
.
(I4)
\tag{I4} x = x_0 + v_0 \left( \frac{v - v_0}{a} \right) + \tfrac12 a \left( \frac{v - v_0}{a} \right)^2.
x
=
x
0
+
v
0
(
a
v
−
v
0
)
+
2
1
a
(
a
v
−
v
0
)
2
.
(
I4
)
It can be simplified into
2
a
(
x
−
x
0
)
=
2
v
0
(
v
−
v
0
)
+
(
v
−
v
0
)
2
=
(
2
v
0
+
v
−
v
0
)
(
v
−
v
0
)
=
(
v
0
+
v
)
(
v
−
v
0
)
.
(I5)
\tag{I5} \begin{array}{rcl} 2a(x - x_0) & = & 2v_0 (v - v_0) + (v - v_0)^2 \newline & = & (2v_0 + v - v_0)(v - v_0) \newline & = & (v_0 + v)(v - v_0). \end{array}
2
a
(
x
−
x
0
)
=
=
=
2
v
0
(
v
−
v
0
)
+
(
v
−
v
0
)
2
(
2
v
0
+
v
−
v
0
)
(
v
−
v
0
)
(
v
0
+
v
)
(
v
−
v
0
)
.
(
I5
)