Two carts with masses of 4.00 kg and 3.40 kg move toward each other on a frictionless track with speeds of 4.70 m/s and 3.56 m/s respectively. The carts stick together after colliding head-on. Find the final speed.

To find the final speed of the two carts after they collide and stick together, you can use the principle of conservation of momentum.

The principle of conservation of momentum states that the total momentum of an isolated system remains constant before and after a collision. In this case, before the collision, the momentum of the first cart is given by the product of its mass (m1 = 4.00 kg) and its initial velocity (v1 = 4.70 m/s): p1 = m1 * v1. Similarly, the momentum of the second cart before the collision is given by the product of its mass (m2 = 3.40 kg) and its initial velocity (v2 = -3.56 m/s) since it's moving in the opposite direction: p2 = m2 * v2.

Since the two carts stick together after colliding, they become one system with a combined mass of m1 + m2 and a final velocity we want to find (vf).

Applying the principle of conservation of momentum, we can set up an equation:

Initial momentum of the system (before collision) = Final momentum of the system (after collision)

p1 + p2 = (m1 + m2) * vf

Substituting the given values:

(4.00 kg * 4.70 m/s) + (3.40 kg * -3.56 m/s) = (4.00 kg + 3.40 kg) * vf

19.6 kg·m/s - 12.1 kg·m/s = 7.4 kg * vf

7.5 kg·m/s = 7.4 kg * vf

Dividing both sides of the equation by 7.4 kg:

vf = 7.5 kg·m/s / 7.4 kg

vf ≈ 1.01 m/s

Therefore, the final speed of the two carts after colliding and sticking together is approximately 1.01 m/s.