Is there a easy way to make a Taylor Expansion in MMa?

Question

We know from special relativity that:$$E^2=m_0^2c^2+p^2c^4$$$$E=\sqrt{m_0^2c^2+p^2c^4}$$$$p=m_0v$$$$E=m_0^2c^2(1+v^2)^{1/2}$$Now I know that I can use a Taylor Series to approximate the square root term when $v\ll c$:$$(1+x)^\alpha=(1+\alpha x)$$So, on paper, I can substitute in $x=v^2$ and $\alpha=\frac{1}{2}$, to get this approximation (when $v \ll c$):$$E=m_0^2c^2(1+\frac{1}{2}v^2)$$$$E=m_0^2c^2+\frac{1}{2}m_0^2c^2v^2$$Is there a clever way that I can do employ a Taylor Series in Mathematica to make the estimation for me, or do I just have to substitute by hand?

Expand[E /. Sqrt[1 + v^2] -> 1 + 1/2 v^2]

Answer 1

Your equation is wrong. It should read (I will write en for energy because E is a reserved symbol):

en^2= (p c)^2 + (m0 c^2)^2
en^2= (m0 v c)^2 +(m0 c^2)^2
en= Sqrt[ (m0 v c)^2 +(m0 c^2)^2]

We expand en in a series of v around zero and simplify:

Simplify[Series[en, {v, 0, 2}], {m0 > 0, c > 0}]