1
2
m
v
2
{\displaystyle {\frac {1}{2}}mv^{2}}
m
{\displaystyle m}
v
{\displaystyle v}
(
γ
−
1
)
m
c
2
{\displaystyle (\gamma -1)mc^{2}}
γ
=
1
1
−
v
2
c
2
{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
lim
c
→
∞
(
γ
−
1
)
m
c
2
=
1
2
m
v
2
{\displaystyle \lim _{c\to \infty }(\gamma -1)mc^{2}={\frac {1}{2}}mv^{2}}
[Solution]
We have to prove that
lim
c
→
∞
(
γ
−
1
)
m
c
2
=
1
2
m
v
2
{\displaystyle \lim _{c\to \infty }(\gamma -1)mc^{2}={\frac {1}{2}}mv^{2}}
=
lim
c
→
∞
(
γ
+
1
)
(
γ
−
1
)
m
c
2
(
γ
+
1
)
{\displaystyle =\lim _{c\to \infty }{\frac {(\gamma +1)(\gamma -1)mc^{2}}{(\gamma +1)}}}
=
lim
c
→
∞
(
γ
2
−
1
2
)
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {(\gamma ^{2}-1^{2})mc^{2}}{\gamma +1}}}
=
lim
c
→
∞
(
γ
)
2
m
c
2
−
(
1
)
2
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {(\gamma )^{2}mc^{2}-(1)^{2}mc^{2}}{\gamma +1}}}
γ
{\displaystyle \gamma }
=
lim
c
→
∞
(
1
1
−
v
2
c
2
)
2
m
c
2
−
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {\mathbf {\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}\right)} ^{2}mc^{2}-mc^{2}}{\gamma +1}}}
=
lim
c
→
∞
m
c
2
(
1
−
v
2
c
2
)
2
−
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {{\frac {mc^{2}}{\mathbf {\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)} ^{2}}}-mc^{2}}{\gamma +1}}}
=
lim
c
→
∞
m
c
2
(
1
−
v
2
c
2
)
−
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {{\frac {mc^{2}}{\mathbf {\left(1-{\frac {v^{2}}{c^{2}}}\right)} }}-mc^{2}}{\gamma +1}}}
=
lim
c
→
∞
m
c
2
c
2
c
2
−
v
2
−
m
c
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {{\frac {mc^{2}c^{2}}{c^{2}-v^{2}}}-mc^{2}}{\gamma +1}}}
=
lim
c
→
∞
m
c
4
c
2
−
v
2
−
(
m
c
2
)
(
c
2
−
v
2
)
c
2
−
v
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {{\frac {mc^{4}}{c^{2}-v^{2}}}{\frac {-(mc^{2})(c^{2}-v^{2})}{c^{2}-v^{2}}}}{\gamma +1}}}
=
lim
c
→
∞
m
c
4
−
m
c
4
+
m
v
2
c
2
c
2
−
v
2
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {\frac {mc^{4}-mc^{4}+mv^{2}c^{2}}{c^{2}-v^{2}}}{\gamma +1}}}
=
lim
c
→
∞
m
v
2
c
2
c
2
(
1
−
v
2
c
2
)
γ
+
1
{\displaystyle =\lim _{c\to \infty }{\frac {\frac {mv^{2}c^{2}}{c^{2}(1-{\frac {v^{2}}{c^{2}}})}}{\gamma +1}}}
=
lim
c
→
∞
m
v
2
(
1
−
v
2
c
2
)
(
γ
+
1
)
{\displaystyle =\lim _{c\to \infty }{\frac {mv^{2}}{(1-{\frac {v^{2}}{c^{2}}})(\gamma +1)}}}
γ
{\displaystyle \gamma }
=
lim
c
→
∞
m
v
2
(
1
−
v
2
c
2
)
(
1
−
v
2
c
2
+
1
)
{\displaystyle =\lim _{c\to \infty }{\frac {mv^{2}}{(1-{\frac {v^{2}}{c^{2}}})({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}+1)}}}
∞
{\displaystyle \infty }
=
m
v
2
(
1
−
v
2
∞
2
)
(
1
−
v
2
∞
2
+
1
)
{\displaystyle ={\frac {mv^{2}}{(1-{\frac {v^{2}}{\infty ^{2}}})({\sqrt {1-{\frac {v^{2}}{\infty ^{2}}}}}+1)}}}
=
m
v
2
(
1
−
0
)
(
1
−
0
+
1
)
{\displaystyle ={\frac {mv^{2}}{(1-0)({\sqrt {1-0}}+1)}}}
=
1
2
m
v
2
{\displaystyle ={\frac {1}{2}}mv^{2}}
Q
.
E
.
D
.
{\displaystyle Q.E.D.}
Ofey 18:58, 24 October 2008 (UTC) ][ reply ]
Sae1962 (discuss • contribs ) 10:03, 30 June 2015 (UTC) [ reply ]
