In a lab experiment, a 1.5-kg cart traveling at 4.3 m/s in the +x direction collides with a 2.5-kg cart traveling at 2.7 m/s in the -x direction. After the collision, you find that the 2.5-kg cart is traveling at 1.9 m/s in the +x direction. How fast and in what direction is the 1.5-kg cart traveling after the collision?

To solve this problem, we can use the principle of conservation of momentum. The law of conservation of momentum states that the total momentum of a closed system remains constant before and after a collision, provided no external forces act on the system.

We can start by calculating the initial momentum of both carts separately. The momentum of an object is defined as the product of its mass and velocity:

Momentum = mass * velocity

For the 1.5-kg cart:
Initial momentum of the 1.5-kg cart = (mass of the 1.5-kg cart) * (velocity of the 1.5-kg cart)
= 1.5 kg * 4.3 m/s
= 6.45 kg·m/s

For the 2.5-kg cart:
Initial momentum of the 2.5-kg cart = (mass of the 2.5-kg cart) * (velocity of the 2.5-kg cart)
= 2.5 kg * (-2.7 m/s) [Since the cart is traveling in the -x direction]
= -6.75 kg·m/s

Now, let's find the final momentum of each cart based on the given information. The final momentum is also equal to the product of mass and velocity after the collision.

For the 2.5-kg cart:
Final momentum of the 2.5-kg cart = (mass of the 2.5-kg cart) * (velocity of the 2.5-kg cart)
= 2.5 kg * 1.9 m/s [Since the cart is traveling in the +x direction after the collision]
= 4.75 kg·m/s

To find the final momentum of the 1.5-kg cart, we subtract the final momentum of the 2.5-kg cart from the initial momentum of the system:

Final momentum of the 1.5-kg cart = (Initial momentum of the system) - (Final momentum of the 2.5-kg cart)
= (Initial momentum of the 1.5-kg cart + Initial momentum of the 2.5-kg cart) - (Final momentum of the 2.5-kg cart)
= (6.45 kg·m/s + (-6.75 kg·m/s)) - (4.75 kg·m/s)
= -4.05 kg·m/s

Since both carts move in the +x direction after the collision, the final momentum of the 1.5-kg cart is also in the positive x-direction.

To find the final velocity of the 1.5-kg cart, we divide the final momentum by its mass:

Final velocity of the 1.5-kg cart = (Final momentum of the 1.5-kg cart) / (mass of the 1.5-kg cart)
= (-4.05 kg·m/s) / (1.5 kg)
≈ -2.70 m/s

So, the 1.5-kg cart is traveling at approximately 2.70 m/s in the -x direction after the collision.