A 10kg mass, M_1, slides to the right across a frictionless floor at v_0 = 5m/s.

Question

A 10kg mass, M_1, slides to the right across a frictionless floor at v_0 = 5m/s. It impacts a massless spring of length 1m with k = 250N/m attached to another 10kg mass, M_2. M_1 sticks to the spring, which is stuck to M_2, and the whole system goes moving along. (a) What is the momentum of the mass-spring-mass system before the collision? What is the mechanical energy of the mass-spring-mass system before the collision? (b) What is the momentum of the mass-spring-mass system after the collision? What is the mechanical energy of the mass-spring-mass system after the collision? Assume that no energy is lost as M_1 attaches to the spring and remember to consider all of the mechanical energy in the system. (c) Is this collision elastic or inelastic? (d) What is the momentum of the mass-spring-mass system as the spring is at its maximum compression? How do you account for this? (e) What is the mechanical energy of the mass spring-mass system as the spring is at its maximum compression? (f) What is the maximum distance to which the spring will be compressed? (g) After the collision, where, between M_1 and M_2, is the center of mass of the system located? (h) (Bonus!) What are the equations of motion for M_1 and M_2 after the collision?

Explanation / Answer

Q1.

Part (a)

Momentum of mass-spring – mass system before collision = M1 Vo = 10*5 = 50 kg-m/s

Mechanical energy of mass-spring – mass system before collision = (1/2)M1 Vo^2 = (1/2) * 10 * 5^2

                                                                                                                        =125 J

Part (b)

Momentum of mass-spring – mass system after collision = Momentum of mass-spring – mass system before collision = 50 kg-m/s

Mechanical energy of mass-spring – mass system after collision = Mechanical energy of mass-spring – mass system before collision = 125 J

Part ( c )

As mechanical energy of system is conserved , and potential energy before and after collision is zero.

So kinetic energy of system is conserved, hence collision is elastic.

Part ( d)

As there is no net external force acting on the system, so momentum of the system is conserved.

So, required momentum of system is = 50 kg-m/s

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