- #1
Kumar8434
- 121
- 5
I tried to get a relativistically correct expression of ##\int_0^t\frac{dp}{ds}dt## similar to the derivation of relativistic energy expression but I got a result which is not defined:
$$\int_0^t\frac{dp}{ds}dt=\int_0^v\frac{d\left(\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\right)}{v}$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v-\int_0^v\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}d\left(\frac{1}{v}\right)$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v-\int_0^v\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\frac{-1}{v^2}dv$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v+\int_0^v\frac{m}{v\sqrt{1-\frac{v^2}{c^2}}}dv$$
Now, the second term is:
$$I=\int_0^v\frac{m}{v\sqrt{1-\frac{v^2}{c^2}}}dv$$
$$=mc\int_0^v\frac{dv}{v\sqrt{c^2-v^2}}$$
Putting ##v=c\sin{x}## and ##dv=c\cos{x}dx##, we get:
$$I=mc\int_0^v\frac{c\cos{x}dx}{c^2\sin{x}\cos{x}}$$
$$=m\int_0^v\csc{x}dx$$
$$=m|\log{|\csc{x}-\cot{x}|}|_0^v$$
$$=m\left|\log{\left|\csc{sin^{-1}\left(\frac{v}{c}\right)}-\cot{sin^{-1}\left(\frac{v}{c}\right)}\right|}\right|_0^v$$
$$=m\left|\log{\left|\frac{c-c \sqrt{1-\frac{v^2}{c^2}}}{v}\right|}\right|_0^v$$
But, this expression is not defined at the lower limit ##v=0##. Have I done something wrong in the maths or does the relativistically correct expression of ##\int_0^t\frac{dp}{ds}dt## not exist?
$$\int_0^t\frac{dp}{ds}dt=\int_0^v\frac{d\left(\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\right)}{v}$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v-\int_0^v\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}d\left(\frac{1}{v}\right)$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v-\int_0^v\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\frac{-1}{v^2}dv$$
$$=\left|\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\right|_0^v+\int_0^v\frac{m}{v\sqrt{1-\frac{v^2}{c^2}}}dv$$
Now, the second term is:
$$I=\int_0^v\frac{m}{v\sqrt{1-\frac{v^2}{c^2}}}dv$$
$$=mc\int_0^v\frac{dv}{v\sqrt{c^2-v^2}}$$
Putting ##v=c\sin{x}## and ##dv=c\cos{x}dx##, we get:
$$I=mc\int_0^v\frac{c\cos{x}dx}{c^2\sin{x}\cos{x}}$$
$$=m\int_0^v\csc{x}dx$$
$$=m|\log{|\csc{x}-\cot{x}|}|_0^v$$
$$=m\left|\log{\left|\csc{sin^{-1}\left(\frac{v}{c}\right)}-\cot{sin^{-1}\left(\frac{v}{c}\right)}\right|}\right|_0^v$$
$$=m\left|\log{\left|\frac{c-c \sqrt{1-\frac{v^2}{c^2}}}{v}\right|}\right|_0^v$$
But, this expression is not defined at the lower limit ##v=0##. Have I done something wrong in the maths or does the relativistically correct expression of ##\int_0^t\frac{dp}{ds}dt## not exist?
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