A block of mass m1 = 2.4 kg slides along a frictionless table with a speed of 8 m/s. Directly in front of it, and moving in the same direction, is a block of mass m2 = 5.0 kg moving at 2.8 m/s. A massless spring with spring constant k = 1140 N/m is attached to the near side of m2, as shown in Fig. 11-38. When the blocks collide, what is the maximum compression of the spring? (Hint: At the moment of maximum compression of the spring, the two blocks move as one. Find the velocity by noting that the collision is completely inelastic to this point.)

To find the maximum compression of the spring, we need to analyze the conservation of momentum and the conservation of energy in the system.

Step 1: Analyzing the collision
- Since the blocks collide and stick together, the collision is completely inelastic. This means that the two blocks move as one after the collision.
- We can use the conservation of momentum to find the velocity of the combined blocks after the collision.
- The initial momentum before the collision is given by m1*v1 + m2*v2, where m1 and m2 are the masses of the blocks, and v1 and v2 are their initial velocities.
- The final momentum after the collision is given by the mass of the combined blocks (m1 + m2) multiplied by their final velocity (let's call it v_final).

Step 2: Finding the velocity of the combined blocks
- Using the conservation of momentum, we equate the initial momentum to the final momentum: m1*v1 + m2*v2 = (m1 + m2)*v_final.
- Plug in the given values: (2.4 kg)(8 m/s) + (5.0 kg)(2.8 m/s) = (2.4 kg + 5.0 kg)*v_final.
- Solve for v_final: v_final = [(2.4 kg)(8 m/s) + (5.0 kg)(2.8 m/s)] / (2.4 kg + 5.0 kg).

Step 3: Analyzing the spring compression
- Once the blocks collide, they compress the spring until it reaches its maximum compression.
- At the moment of maximum compression, the blocks move as one, and all their initial kinetic energy is transferred into the potential energy stored in the compressed spring.
- We can use the conservation of energy to find the maximum compression of the spring.
- The initial kinetic energy is given by (1/2)*m_total*v_final^2, where m_total is the sum of the masses of the blocks (m1 + m2), and v_final is the velocity of the combined blocks after the collision.
- The potential energy stored in the compressed spring is given by (1/2)*k*x^2, where k is the spring constant and x is the compression of the spring.
- Equate the initial kinetic energy to the potential energy: (1/2)*m_total*v_final^2 = (1/2)*k*x^2.
- Plug in the given values: (1/2)*(2.4 kg + 5.0 kg)*(v_final)^2 = (1/2)*(1140 N/m)*x^2.
- Solve for x (the maximum compression of the spring): x = sqrt([(2.4 kg + 5.0 kg)*(v_final)^2] / (1140 N/m)).

Step 4: Calculate the maximum compression
- Substitute the value of v_final from Step 2 into the equation for x: x = sqrt([(2.4 kg + 5.0 kg)*v_final^2] / (1140 N/m)).
- Calculate the value of x using the given masses and velocities, and the calculated v_final.

This will give you the maximum compression of the spring.