two air carts of mass m1=0.87kg and m2=0.48kg are placed on a frictionless track.cart 1 is at rest initally, and a spring bumper with a force constant of 690N/m. cart 2 has a flat metal surface for a bumper, and move toward the bumper of the stationary cart with an intial speed v=0.69m/s. Assume that positive x-axis is directed toward the direction of motion of cart 2.

a)what is the speed of the two cart at the moment when their speeds are equal?
b) how much energy is stored in the spring bumper?

a) To find the speed of the two carts when their speeds are equal, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.

1. Find the initial momentum of each cart:
- Momentum of cart 1 (m1) before the collision = 0 (as it is at rest initially)
- Momentum of cart 2 (m2) before the collision = m2 * v (where v is the initial speed of cart 2)

2. Find the final momentum of each cart:
- Momentum of cart 1 (m1) after the collision = m1 * v1 (where v1 is the final speed of cart 1)
- Momentum of cart 2 (m2) after the collision = m2 * v2 (where v2 is the final speed of cart 2)

3. Apply conservation of momentum:
- m1 * v1 + m2 * v2 = m2 * v

4. Solve for v1 (since we want to find the speed of cart 1):
- v1 = (m2 * v - m2 * v2) / m1

Now we have an equation for v1 in terms of v2. To find the value of v2, we need to consider the conservation of energy.

b) To find the energy stored in the spring bumper, we can use the principle of conservation of energy. The initial kinetic energy of cart 2 will be converted into potential energy stored in the spring bumper.

1. Find the initial kinetic energy of cart 2:
- Initial kinetic energy of cart 2 = (1/2) * m2 * v^2

2. Find the potential energy stored in the spring bumper, which is equal to the work done on the spring:
- Potential energy stored in the spring bumper = (1/2) * k * x^2
(where k is the force constant of the spring bumper, and x is the displacement of the spring from its equilibrium position)

Note: The displacement of the spring can be calculated by finding the compression distance of the spring when cart 2 collides with cart 1.