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Kinetic energy of any particle (who has mass) is $$T=\frac{1}{2} m\ddot{x}^2$$ OP had took potential as potential energy, that was wrong. $$U=-\int \vec F\cdot d\vec l $$ For the case, The forc...
Answer
#1: Initial revision
Kinetic energy of any particle (who has mass) is $$T=\frac{1}{2} m\ddot{x}^2$$
OP had took potential as potential energy, that was wrong.
$$U=-\int \vec F\cdot d\vec l
$$
For the case, The force was $$F=\frac{1}{4\pi\epsilon_0} \frac{Qq}{r^2}\hat r$$
So the lagrangian is $$L=\frac{1}{2}m\dot{x}^2-\frac{1}{4\pi\epsilon_0} \frac{Qq}{r}$$
Now you can get a satisfied answer.