Motion of Charged Particle in Uniform Electric Field (Longitudinal)

Consider a charged particle of mass m and charge q in uniform electric field of strength E.
    The force experienced by charge q is F4 = qE                        … (1)
    The acceleration in the direction of force is
        A = d²r/dr² = qE/m                                … (2)

Now, integrating Eq. (2), velocity of the particle is
            dr/dt = v= q/E/m. t + C1
where C1 is constant of integration.
Let u be the initial velocity of particle at t =0.        . : C1 = u
            dr./dt = v = qE/m . t + u                            … (3)

Displacement of the particle is obtained by integrating Eq. (3).
            R = qE/m t²/2 + ut + C₂

Where C₂ is another constant of integration.
Let at t = 0, initial position r = r₀. Thus C₂ = r0

. : r = qE/m t²/2 + ut + r₀                                         … (4)
Or displacement
     s = r = r₀ = ut + ½ qE/m t²                                     … (5)


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