The following text is a method for finding the magnetic field as described in Purcell's Electricity and Magnetism (page $151$, the top part). Measure the force on the particle when its velocity is $\bf{v}$; repeat with $\bf{v}$ in some other direction. Now find a $\bf{B}$ that will make $\tex...

Consider a ball that strikes a stationary box and has momentum $mv$. If the ball rebounds with the same speed it had before the collision, its momentum is $-mv$. Therefore, the box must have a momentum of $2mv$ in order for momentum to be conserved. However this would mean the kinetic energy afte...