Two air-track carts are sliding to the right at 1.0 m/s and held together by a string. The spring between them has a spring constant of 140 N/m and is compressed 4.5 cm. The carts slide past a flame that burns through the string holding them together. Afterward, what are the speed and direction of each cart?

To determine the speed and direction of each cart after the string is burned, we need to apply the principle of conservation of momentum.

Conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, before the string is burned, the two carts are moving together as a system, and there are no external forces acting on them.

Let's denote the mass of the first cart as m1 and the mass of the second cart as m2. Also, let's denote the initial velocity of the first cart as v1 and the initial velocity of the second cart as v2.

The total momentum before the string is burned can be calculated as:
Initial total momentum = m1 * v1 + m2 * v2

After the string is burned, the carts separate and move independently. Let's denote the final velocity of the first cart as vf1 and the final velocity of the second cart as vf2.

According to conservation of momentum, the total momentum after the string is burned must also be equal to the initial total momentum:
Final total momentum = m1 * vf1 + m2 * vf2 = Initial total momentum

We know that the carts were moving together to the right before the string was burned, so their initial velocities are v1 = 1.0 m/s and v2 = 1.0 m/s.

Now we need to determine the masses of the carts. Since we don't have that information, we can't determine the exact values for vf1 and vf2. However, we can use the principle of conservation of kinetic energy to calculate the final velocities in terms of the initial velocities and spring compression.

When the carts were held together by the compressed spring, they had potential energy stored in the spring. After the string is burned and the spring is released, this potential energy is converted to kinetic energy. The conservation of kinetic energy states that the total kinetic energy of a system remains constant if no external forces act on it.

The initial kinetic energy of the system can be calculated as:
Initial kinetic energy = 0.5 * m1 * v1^2 + 0.5 * m2 * v2^2

The final kinetic energy of the system after the string is burned can be calculated as:
Final kinetic energy = 0.5 * m1 * vf1^2 + 0.5 * m2 * vf2^2

Since the spring constant and compression distance are given, we can calculate the initial potential energy stored in the spring and equate it to the final kinetic energy of the system.

The initial potential energy stored in the spring can be calculated using Hooke's Law:
Initial potential energy = 0.5 * k * x^2
where k is the spring constant (140 N/m) and x is the compression distance (4.5 cm or 0.045 m).

The final kinetic energy of the system can be calculated as:
Final kinetic energy = 0.5 * m1 * vf1^2 + 0.5 * m2 * vf2^2

Setting the initial potential energy equal to the final kinetic energy, we have:
0.5 * k * x^2 = 0.5 * m1 * vf1^2 + 0.5 * m2 * vf2^2

Now, you need to know the masses of the carts to calculate the final velocities vf1 and vf2. Once you have the masses, you can solve the equation above to find the final velocities. Remember that the carts can move in either direction (positive or negative) depending on the chosen reference frame.