Proton in uniform electric field - time to travel distance d

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Consider a proton with mass m and charge e, initially at rest in a uniform electric field E. We want to determine the time it takes for the proton to travel a distance d in this field. Let's visualize the setup: the proton is placed in a uniform electric field pointing to the right. The proton will accelerate due to the electric force acting on it. The electric force acting on the proton is given by F equals e times E. According to Newton's second law, force equals mass times acceleration. Equating these two expressions, we get eE equals ma. Solving for acceleration, we find a equals eE divided by m. This acceleration is constant since the electric field is uniform. The proton starts from rest, so its initial velocity u is zero. We know its constant acceleration a is eE divided by m. Using the kinematic equation for distance: d equals ut plus one-half a t squared. Substituting our known values: d equals zero plus one-half times eE over m times t squared. We can now solve this equation for the time t. Starting with our equation: d equals one-half times eE over m times t squared. To solve for t, we first multiply both sides by 2m over eE. This gives us: 2md over eE equals t squared. Taking the square root of both sides, we find the time t equals the square root of 2md over eE. This is our final answer for the time taken by the proton to travel distance d in the electric field.