INTRO:A proton and an alpha pwrite-up are momentarily at rest at a distance r from each various other. They then begin to relocate apart. Find the speed of the proton by the time the distance in between the proton and the alpha ppost doubles.Both pposts are positively charged. The charge and also the mass of the proton are, respectively, e and also m. The charge and also the mass of the alpha ppost are, respectively, 2e and 4m.PROBLEM-SOLVING STRATEGY 25.1 Conservation of energy in charge interactionsMODEL: Check whether tright here are any dissipative forces that would certainly proccasion the mechanical energy from being conoffered.VISUALIZE: Draw a before-and-after photographic depiction. Define symbols that will certainly be provided in the difficulty, list recognized worths, and also determine what you are trying to find.SOLVE: The mathematical representation is based on the regulation of conservation of mechanical energy:

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Is the electrical potential provided in the difficulty statement? If not, you"ll should use a well-known potential, such as that of a point charge, or calculate the potential using the procedure provided in Problem-Solving Strategy 25.2.KfandKiare the sums of kinetic energies of all moving pwrite-ups.Some difficulties may require added conservation laws, such as conservation of charge or conservation of momentum.ASSESS: Check that your outcome has actually the correct systems and also substantial numbers, is reasonable, and answers the question.ModelThe pshort articles described in this trouble connect under the effect of the electric force, which is a conservative force, so the system"s mechanical power is conoffered.Visualize-----------------------------------------------------------------------------------------------------PART A:Which of the complying with amounts are unknown?A. initial separation of the particlesB. last separation of the particlesC. initial rate of the protonD. initial speed of the alpha particleE. final rate of the protonF. final speed of the alpha particleG. mass of the protonH. mass of the alpha particleI. charge of the protonJ. charge of the alpha particle
Get in the letters of all the correct answers in alphabetical order. Do not usage commas. For instance, if A, C, and D are unknowns, enter ACD.SOLUTION:The last separation in between the pposts is, fundamentally, known: It is twice the initial separation, or 2r. In this trouble, then, there are really only two unknowns: the last speed of the proton, (vf)p, which is what you are trying to uncover, and the last rate of the alpha pwrite-up, (vf)α.Respeak to that a problem with two unknowns calls for two equations to be addressed. Here, the regulation of conservation of mechanical power offers one of the equations. To discover the second equation, think what other physical quantity besides energy is conserved, and also translate that into a mathematical expression. But before you perform that, it"s valuable if you finish a before-and-after photographic depiction of the difficulty. Your drawing might look choose this:

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answer: BEF-----------------------------------------------------------------------------------------------------PART B:Find the rate of the proton (vf)pby the time the distance between the pposts doubles.Expush your answer in regards to some or all of the quantities e, m, r, and also ϵ0.SOLUTION:Using the regulation of conservation of mechanical energy, we know that the amount of the initial energies is equal to the sum of the final energies.At initially, once the two pwrite-ups are right beside each various other, tright here is no kinetic energy yet, however rather a little bit of potential energy. The potential energy can be composed utilizing U = qV ->Uelec= 1/<4π⋅ϵ0>⋅q1q2/distAs such, because initial dist = r,q1= e & q2= 2eThe initial potential energy =Ui= 1/<4π⋅ϵ0>⋅2e⋅e/r = 1/<4π⋅ϵ0>⋅2e2/r