$2.3.39.$ One research institute decided to use an expression for the potential energy of point charges in the form $U' = \frac{kqQ}{r} - \frac{kqQ}{R}$, where $R$ is a constant distance set once and for all. Will using $U'$ instead of $U = \frac{kqQ}{r}$ affect the results of particle motion calculations?
Expression for the potential energy $$ U = \frac{kqQ}{r} $$ $$ U' = \frac{kqQ}{r} - \frac{kqQ}{R} $$ By statement, $R$ is a constant distance $$ R = \text{const} $$ Derivatives for potential energy of expressions $$ (U_x)' = p $$ $$ (U)' = - \frac{kqQ}{r^2} \, dr $$ $$ (U')' = - \frac{kqQ}{r^2} \, dr $$ The derivative of the constant is zero $$ (\text{const})' = 0 \Rightarrow (R)' = 0 $$ $$ (U)' = (U')'! $$ Thus, the speeds remain unchanged and the motion of the particles is not affected $$ V = V'! $$
$ \text{No, it won't!} $
Almaskhan Arsen