Equations are in image: Let's explore the idea that momentum is mass times velocity. As in the textbook, let $m$

represent the rest mass $($sometimes labeled $m_0$ but here we will just use $m).$

a$)$ Expand the energy $E$ in Eq$.(3\mathrm{.}41)$ in a Taylor series in $v^2$ and verify that the

result is Eq$.(3\mathrm{.}44).$

b$)$ Use the expression for the relativistic invariant $p_{}{p}^{}$ in Eq$.(3\mathrm{.}43)$ to solve for

$E$ in terms of $p=|vec(p)|.$ Expand this in a Taylor series in $p^2$ to order $p^4.($This is

basically problem $5\mathrm{.}4$ in the textbook.$)$

c$)$ As you will notice, $T=E-m{c}^2$ from these two approaches is not the same if

you assume $p=mv.$ Instead, one must use $p=mv.$ Take the magnitude of

the left side of Eq$.(3\mathrm{.}47)$ and expand $$ as a Taylor series in $v^2.$ Then substitute

the resulting expression for $p$ into your expansion for $T$ in $($b$)$ above, keeping

terms up to order $v^4.$

d$)$ What does this tell you?