An electron and a proton, starting from rest, are accelerated through an electric potential difference of the same magnitude. in the process, the electron acquires a speed ve, while the proton acquires a speed vp. find the ratio ve/vp.

The charges are the same in absolute value, so the change of potential energy is the same. That means that the change in kinetic energy is also the same. Then:

1 = Ke/Kp = m_e *v_e^2 / m_p * v_p^2, or

v_e/v_p = sqrt( m_p/m_e),

So the speed of the electron will be sqrt( m_p/m_e) times greater than the speed of the proton

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