ProblemA newly discovered light positively charged particle has a mass of m and charge q. Suppose it moves within the vicinity of an extremelyheavy (fixed in place) particle with a positive charge Q and mass M. When the light particle is xi distance from the heavy particle, it is movingdirectly away from the heavy particle with a speed of vi. a) What is the lighter particle's speed when it is xf away from the heavy particle?(Consider the Newtonian Gravitation acting between the two charged particles. Ignore the effects of external forces)Solution:We may solve this using two approaches. One involves the Newton's Laws and the other involving Work-Energy theorem.To avoid the complexity of vector solution, we will instead employ the Work-Energy theorem, more specifically, the Conservation of EnergyPrinciple.Let us first name the lighter particle as object 1 and the heavy particle as object 2.Through work-energy theorem, we will take into account all of the energy of the two-charged particle system before and after traveling acertain distance asKE16+ KE2F + PENewtonianf + Uelasticf + Uelectricf = KE11 + KE2i + PENewtoniani ++ UelectriciSince the heavy particle remains fixed, before and after the motion of the lighter particle, it does not have any velocity, moreover, there isno spring in involved, soKE14 ++ Uelectricf =+ Uelectrici(Equation 1)For all energies, we know the following1KE==muGm,m2PENewtonian1kx²Uelastic=Velectric = (1//(r)) Gm,m2PENewtonianUelastic = kx?Velectric = (1/where in we havem1 = m, m2 = M, q1 = q and q2 = QBy substituting all these to Equation 1 and then simplifying results to= sqrt v2 + ( (VQm)) - (1/x) ) +

eavy (fixed in place) particle with a positive charge Q and mass M. When the light particle is xi distance from the heavy particle, it is movin directly away from the heavy particle with a speed of vi. a) What is the lighter particle's speed when it is xf away from the heavy particle? Consider the Newtonian Gravitation acting between the two charged particles. Ignore the effects of external forces) Colution: Ve may solve this using two approaches. One involves the Newton's Laws and the other involving Work-Energy theorem. o avoid the complexity of vector solution, we will instead employ the Work-Energy theorem, more specifically, the Conservation of Energy Principle. et us first name the lighter particle as object 1 and the heavy particle as object 2. hrough work-energy theorem, we will take into account all of the energy of the two-charged particle system before and after traveling a ertain distance as KE1F+ KE2F + PENewtonianf + Uelasticf + Uelectricf = KE1i + KE2i + PENewtoniani + + Uelectrici since the heavy particle remains fixed, before and after the motion of the lighter particle, it does not have any velocity, moreover, there is o spring in involved, so KE11+ + Uelectricf = + Uelectrici (Equation 1) For all energies, we know the following KE E=mv Gm,m2 PENewtonian Uelastic Velectric = (1/

eavy (fixed in place) particle with a positive charge Q and mass M. When the light particle is xi distance from the heavy particle, it is movin directly away from the heavy particle with a speed of vi. a) What is the lighter particle's speed when it is xf away from the heavy particle? Consider the Newtonian Gravitation acting between the two charged particles. Ignore the effects of external forces) Colution: Ve may solve this using two approaches. One involves the Newton's Laws and the other involving Work-Energy theorem. o avoid the complexity of vector solution, we will instead employ the Work-Energy theorem, more specifically, the Conservation of Energy Principle. et us first name the lighter particle as object 1 and the heavy particle as object 2. hrough work-energy theorem, we will take into account all of the energy of the two-charged particle system before and after traveling a ertain distance as KE1F+ KE2F + PENewtonianf + Uelasticf + Uelectricf = KE1i + KE2i + PENewtoniani + + Uelectrici since the heavy particle remains fixed, before and after the motion of the lighter particle, it does not have any velocity, moreover, there is o spring in involved, so KE11+ + Uelectricf = + Uelectrici (Equation 1) For all energies, we know the following KE E=mv Gm,m2 PENewtonian Uelastic Velectric = (1/