This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Hey AUDREY, good to see you again. Here's the solution for the collision question. Given: Mass of sphere A, m_A = m Mass of sphere B, m_B = 2m Initial velocity of sphere A, u_A = u Initial velocity of sphere B, u_B = 0 (at rest) Coefficient of restitution, e = (1)/(2) a) The speed of each sphere after impact Step 1: Apply the principle of conservation of linear momentum. m_A u_A + m_B u_B = m_A v_A + m_B v_B Substitute the given values: m(u) + 2m(0) = m v_A + 2m v_B mu = m v_A + 2m v_B Divide by m: u = v_A + 2v_B (1) Step 2: Apply the definition of the coefficient of restitution. e = (v_B - v_A)/(u_A - u_B) Substitute the given values: (1)/(2) = (v_B - v_A)/(u - 0) (u)/(2) = v_B - v_A (2) Step 3: Solve equations (1) and (2) simultaneously for v_A and v_B. From (2), v_A = v_B - (u)/(2). Substitute this into (1): u = (v_B - (u)/(2)) + 2v_B u = 3v_B - (u)/(2) u + (u)/(2) = 3v_B (3u)/(2) = 3v_B v_B = (u)/(2) Now substitute v_B back into (2) to find v_A: (u)/(2) = (u)/(2) - v_A v_A = 0 The speeds of the spheres after impact are: Speed of sphere A: v_A = 0 Speed of sphere B: v_B = (u)/(2) b) The magnitude of the impulse exerted by A on B Step 1: Calculate the impulse using the change in momentum of sphere B. J = m_B v_B - m_B u_B Substitute the values: J = (2m)((u)/(2)) - (2m)(0) J = mu - 0 J = mu The magnitude of the impulse exerted by A on B is mu. c) The loss of kinetic energy due to the impact Step 1: Calculate the initial total kinetic energy. KE_initial = (1)/(2) m_A u_A^2 + (1)/(2) m_B u_B^2 KE_initial = (1)/(2) m (u)^2 + (1)/(2) (2m) (0)^2 KE_initial = (1)/(2) mu^2 Step 2: Calculate the final total kinetic energy. KE_final = (1)/(2) m_A v_A^2 + (1)/(2) m_B v_B^2 KE_final = (1)/(2) m (0)^2 + (1)/(2) (2m) ((u)/(2))^2 KE_final = 0 + (1)/(2) (2m) ((u^2)/(4)) KE_final = m (u^2)/(4) = (1)/(4) mu^2 Step 3: Calculate the loss of kinetic energy. Loss = KE_initial - KE_final Loss = (1)/(2) mu^2 - (1)/(4) mu^2 Loss = ((1)/(2) - (1)/(4)) mu^2 Loss = (1)/(4) mu^2 The loss of kinetic energy due to the impact is (1)/(4) mu^2. Send me the next one 📸