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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You're on a roll — here are the solutions for questions 46-49. 46. The law of conservation of momentum applies to a system of colliding objects if there are no net external forces acting on the system. If there are no net external forces, then the net external impulse (which is the change in momentum due to external forces) is zero. Answer: (D) the net external impulse is zero 47. Step 1: Calculate the time for the first fall (t_1). The ball falls from rest (v_0 = 0) from a height h_1 = 4.9 \, m. Using the kinematic equation h = v_0 t + (1)/(2)gt^2: 4.9 = 0 · t_1 + (1)/(2)(9.8)t_1^2 4.9 = 4.9t_1^2 t_1^2 = 1 t_1 = 1 \, s Step 2: Calculate the velocity just before the first impact (v_1). Using v = v_0 + gt: v_1 = 0 + (9.8)(1) = 9.8 \, m/s Step 3: Calculate the velocity just after the first impact (v_1'). The coefficient of restitution e = 0.75 is given by e = (v_1')/(v_1). v_1' = e · v_1 = 0.75 · 9.8 \, m/s = 7.35 \, m/s This is the initial upward velocity for the second bounce. Step 4: Calculate the time for the second flight (up and down, t_2). The time to reach the peak height after bouncing is t_up = (v_1')/(g). t_up = 7.35 \, m/s9.8 \, m/s^2 = 0.75 \, s The total time for the ball to go up and come back down to the surface for the second hit is t_2 = 2 · t_up. t_2 = 2 · 0.75 \, s = 1.5 \, s Step 5: Calculate the total time until the second hit. Total time = t_1 + t_2 = 1 \, s + 1.5 \, s = 2.5 \, s. Answer: (C) 2.5 sec 48. Step 1: Write the formulas for kinetic energy and momentum. Kinetic energy: KE = (1)/(2)mv^2 Momentum: p = mv Step 2: Express kinetic energy in terms of momentum. From p = mv, we get v = (p)/(m). Substitute this into the kinetic energy formula: KE = (1)/(2)m ((p)/(m))^2 = (1)/(2)m (p^2)/(m^2) = (p^2)/(2m) Step 3: Use the condition that kinetic energies are equal (KE_1 = KE_2). (p_1^2)/(2m_1) = (p_2^2)/(2m_2) Step 4: Solve for the ratio of momenta. (p_1^2)/(m_1) = (p_2^2)/(m_2) (p_1^2)/(p_2^2) = (m_1)/(m_2) Taking the square root of both sides: (p_1)/(p_2) = sqrt((m_1)/(m_2)) So, the ratio of their momentum is sqrt(m_1) : sqrt(m_2). Answer: (A) sqrt(m_1) : sqrt(m_2) 49. An inelastic collision is characterized by the conservation of momentum, but the kinetic energy of the system is not conserved. Some kinetic energy is converted into other forms of energy, such as heat, sound, or deformation. Answer: (D) momentum is conserved but kinetic energy is not conserved What's next?