Speed and Velocity of Carts in Collision on Frictionless Track

What is the speed of block 1 (in the lab frame) when the spring is maximally compressed?

Can you explain the step-by-step process of finding the speed of block 1 in the lab frame and why it differs from the answer in the lab frame?

Answer:

The speed of block 1 (cart 1) in the lab frame when the spring is maximally compressed and the velocity of cart 1 in the lab frame after the collision can be found by applying the principles of conservation of momentum and kinetic energy.

Problem Analysis: We have two carts moving towards each other on a frictionless track. Cart 1 has a mass of 3.5 kg and an initial velocity (Vi) of 4 m/s. Cart 2 has a mass of 8 kg and an initial velocity (Vi) of 0 m/s. A spring with a spring constant of 300 N/m is placed between the carts, and the collision between the carts is elastic.

Step-by-Step Solution: First, let's find the velocity of block 1 (cart 1) in the lab frame when the spring is maximally compressed. Since the collision is elastic, both momentum and kinetic energy are conserved.

Using the conservation of momentum and kinetic energy equations, we can calculate the final velocity of cart 1 in the lab frame. The final velocities will depend on the masses and initial velocities of both carts. Solve these equations to find the required velocities.