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MathematicalPhysicist
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Homework Statement
The question as follows:
Calculate the energy that is required to accelerate a particle of rest mass ##m\ne 0## from speed ##v## to speed ##v+\delta v## (##\delta v \ll v##).
Show that it would take an infinite amount of energy to accelerate the particle to the speed of light.
Homework Equations
The Attempt at a Solution
Here's what I have done so far, the 4-momentum before is ##(m,mv)## and the 4-momentum after is: ##(E,m(v+\delta v))##, the square of the 4 momentum is conserved, i.e:
$$E^2 - m^2(v+\delta v)^2 = m^2-m^2v^2$$
After rearranging I get the following equation for the energy:
$$E=mv\sqrt{1/v^2+2\delta v /v +(\delta v/v)^2}$$
I think I have a mistake somewhere, since I don't know how to expand this in a Taylor series, I have the expansion ##\sqrt{1+x} \approx 1+1/2 x##, but here I have ##1/v^2## inside the sqrt.
Perhaps I am wrong with the 4-momentum or something else, any tips?
Thanks in advance.
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