Final answer:
Using the conservation of momentum, the total initial momentum (5.25 kg·m/s) divided by the total mass after the collision (4.0 kg) gives a final velocity of approximately 1.3125 m/s to the right. Therefore, the final velocity of the combined carts is closest to 1.0 m/s to the right, which corresponds to option B.
Step-by-step explanation:
The final velocity of two carts after a head-on collision can be determined using the principle of conservation of momentum. Since there are no external forces acting on the carts (assuming a frictionless surface), the total momentum of the system before the collision is equal to the total momentum of the system after the collision. The initial momentum of the 2.5 kg cart moving to the right is (2.5 kg × 3.0 m/s = 7.5 kg·m/s). The initial momentum of the 1.5 kg cart moving to the left is (1.5 kg × -1.5 m/s = -2.25 kg·m/s). The total initial momentum is 7.5 kg·m/s - 2.25 kg·m/s = 5.25 kg·m/s. After the collision, the carts stick together, so their combined mass is 2.5 kg + 1.5 kg = 4.0 kg. To find the final velocity, we use the equation:
total initial momentum = total final momentum
5.25 kg·m/s = (4.0 kg) × final velocity.
Solving for the final velocity, we get final velocity = 5.25 kg·m/s / 4.0 kg = 1.3125 m/s. Since the result is positive, the direction is to the right. Therefore, the closest option given is B) 1.0 m/s to the right.