Here are two blocks of mass 1 kg.The rear block moves with a speed of 2 m/s towards the front block kept at rest. The spring attached to front block is light and has a spring constant 50N/m. Find the maximum compression of the spring

To solve this problem, we need to analyze the situation using the principles of conservation of momentum and energy. We have two blocks, where one block (the rear block) is moving towards the other block (the front block) that is at rest. When the moving block collides with the stationary block, it compresses the spring attached to the front block. Our goal is to find the maximum compression of the spring during this interaction.

Step 1: Understanding the Initial Conditions

We have:

• Mass of each block, m = 1 kg
• Initial speed of the rear block, v = 2 m/s
• Spring constant, k = 50 N/m

The front block is initially at rest, so its initial velocity is 0 m/s. When the rear block collides with the front block, it will transfer some of its kinetic energy to the spring, causing it to compress.

Step 2: Applying Conservation of Momentum

Before the collision, the total momentum of the system is given by the moving block:

Initial momentum = m * v = 1 kg * 2 m/s = 2 kg·m/s

After the collision, let’s assume both blocks move together momentarily at a common velocity (v_f) when the spring is maximally compressed. The total mass of the two blocks is 2 kg. Therefore, we can express the final momentum as:

Final momentum = (m + m) * v_f = 2 kg * v_f

Setting the initial momentum equal to the final momentum gives us:

2 kg·m/s = 2 kg * v_f

From this, we find:

v_f = 1 m/s

Step 3: Calculating Kinetic Energy Before and After Compression

Next, we calculate the kinetic energy before the collision and the kinetic energy at maximum compression. The initial kinetic energy (KE_initial) of the moving block is:

KE_initial = 0.5 * m * v^2 = 0.5 * 1 kg * (2 m/s)^2 = 2 J

At maximum compression, the kinetic energy of the system is converted into potential energy stored in the spring (PE_spring) and the kinetic energy of both blocks moving together:

KE_final = 0.5 * (m + m) * v_f^2 = 0.5 * 2 kg * (1 m/s)^2 = 1 J

The potential energy stored in the spring at maximum compression (x) is given by:

PE_spring = 0.5 * k * x^2

Step 4: Setting Up the Energy Conservation Equation

According to the conservation of energy, the initial kinetic energy equals the sum of the kinetic energy at maximum compression and the potential energy in the spring:

KE_initial = KE_final + PE_spring

Substituting the values we calculated:

2 J = 1 J + 0.5 * 50 N/m * x^2

Rearranging gives:

1 J = 0.5 * 50 N/m * x^2

1 J = 25 N/m * x^2

Now, solving for x:

x^2 = 1 J / 25 N/m = 0.04 m²

x = √0.04 m² = 0.2 m

Final Result

The maximum compression of the spring is 0.2 meters or 20 centimeters.

This result illustrates how energy is transferred and transformed during the collision, showcasing the interplay between kinetic and potential energy in a mechanical system.