As shown in the figure, a block of mass 2.0 kg travels at a velocity of 1.0 m/s on a frictionless horizontal surface towards a second block of equal mass that is at rest. The spring attached to the stationary block has a spring constant of 100 N/m . Determine the maximum compression of the spring.

Step-by-step solution

Let m₁ = 2.0 kg , u₁ = 1.0 m/s , m₂ = 2.0 kg , u₂ = 0 m/s , and k = 100 N/m . At the instant of maximum compression, both blocks move with the same common velocity v . By conservation of linear momentum: m₁ u₁ + m₂ u₂ = (m₁ + m₂) v 2.0 1.0 + 0 = (2.0 + 2.0) v 2.0 = 4.0 v v = 0.5 m/s By conservation of mechanical energy, the loss in kinetic energy is stored as the potential energy of the spring: 1 2 m₁ u₁^2 = 1 2 (m₁ + m₂) v^2 + 1 2 k x^2 where x is the maximum compression of the spring. 1 2 2.0 (1.0)^2 = 1 2 4.0 (0