Imagine that George and Evil George are in combat as described in this module’s lecture. Specifically, in the ground frame, let George have velocity $v_x$ and Evil George have velocity $-v_x$. When they jab at each other, have them do so in the $y$-direction with velocities $v_y$ and $-v_y$ respectively. Assume that both George and Evil George time their jousts perfectly, so each weapon hits the other squarely. For ease, assume $v_y \ll v_x$.

From Newtonian physics, we expect the impact of a jab to involve a product of the mass of the weapon times its speed, $m v_y$. Relativity, and the parable of the jousters, inspires us to update this formula in a specific way: we allow the mass of an object to depend on its velocity. Let’s denote this dependence by writing $m_0[v_x]$, where we are using our assumption of $v_y \ll v_x$ to only include $v_x$ dependence, and $m_0[v_x]$ is the function of velocity that we want to determine. Using this notation, the momentum of George’s jab in the $y$-direction is expressed as $m_0[v_x] v_y$. Let’s figure out the explicit form of $m_0[v_x]$ in a manner similar to what we did in lecture, but now comparing observations in the stadium frame and in George’s frame.