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Regard as an electron and a proton at an instant in time to be at a separation of 5 cm from each other. Presume the pair to be totally isolated from its surroundings in the vacuum of outer space.
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By means of Newton's third law the Coulomb force of attraction exerted by each particle on the other is of one and the same magnitude. The electron has greatly less mass than the proton however so the electron experiences a much greater acceleration. The force sprightly determines the acceleration so it governs how fast the velocity will change but it doesn't directly determine what is the velocity. From the information given every particle could have any velocity greater than or equal to 0 but less than the speed of light.
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