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52708A Let two particles of equal mass collide. Particle 1 has initial velocity , directed to the right, and particle 2 is initially stationary. Answer: This type of problems have a common approach, simply the masses and the initial velocities are different. Let us first derive the equations in common. Let the masses of the particles be m1 and m2 and their initial velocities(along line joining) are u1 and u2 respectively (u1 must be > u2 for collision), and after collision moving with velocities v1 and v2 respectively(v2 >v1 ; separation) According to law of conservation of linear momentum and of kinetic energy if the collision is elastic. m1u1 + m2u2 = m1v1 +m2v2 …………(1) and (1/2)m1u12 + (1/2)m2u22 =(1/2)m1v12 + (1/2)m2v22, which gives m1u12 + m2u22 = m1v12 + m2v22 ………(2) To avoid square terms we can solve the equations for final velocities in this way: rearranging the equations m1u1 - m1v1 = m2v2 - m2u2 … ……….(1) and m1u12 - m1v12 = m2v22 - m2u22 …….(2) dividing 2 by 1 and simplifying we get u1 + v1 = u2 + v2 or u1 – u2 (velocity of approach) = v2 – v1 (velocity of separation) ….(3) using equation 1 and 3 is easier. Part A If the collision is elastic, what are the final velocities and of particles 1 and 2? Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express each velocity in terms of . 1) mv + 0 + = mv1 + mv2 gives v = v1 + v2 and 2) v = v2 – v1 adding the two we get v2 = v and by subtracting v1 = 0 Answer: 0, v Part B Now suppose that the collision is perfectly inelastic. What are the velocities and of the two particles after the collision? Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of . If the collision is perfectly inelastic the particles will stick together and have same velocity after collision v1 = v2. We cant use conservation of energy equation and hence using first equation mv = (m + m)v1 gives v1 = v/2 so Answer: v/2, v/2. Part C Now assume that the mass of particle 1 is , while the mass of particle 2 remains the collision is elastic, what are the final velocities and of particles 1 and 2? . If Give the velocity of particle 1 followed by the velocity comma. Express the velocities in terms of . of particle 2, separated by a 1) 2) 2mv + 0 = 2m v1 + mv2 gives 2v1 + v2 = 2v and v = v2 – v1 Solving the two equations we get v1 = v/3 and v2 = 4v/3 Answer: v/3, 4v/3. Part D Let the mass of particle 1 be and the mass of particle 2 be . If the collision is perfectly inelastic, what are the velocities of the two particles after the collision? Give the velocity of particle 1 followed by the velocity of particle 2, separated by a comma. Express the velocities in terms of . 1) mv = mv1 +3mv2 and as perfectly inelastic collision v1 = v2: we get v1 = v2 = v/4 Answer: v/4, v/4