A cart is moving at 1.01 m/s to the left has a head-on collision with a 28.5 kg cart traveling 2.04 m/s to the right. If the velocity to the first cart is 2.99 m/s to the right, what is its mass?

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum of the system before the collision should be equal to the total momentum after the collision.

Momentum is defined as the product of an object's mass and its velocity.

Let's denote:
m1 = mass of the first cart (unknown)
v1 = velocity of the first cart before the collision = 2.99 m/s to the right
m2 = mass of the second cart = 28.5 kg
v2 = velocity of the second cart before the collision = 2.04 m/s to the right

Since the first cart is moving to the left, we can represent its velocity as -2.99 m/s to the right.

The momentum before the collision can be calculated as:
Initial momentum of the first cart = m1 * v1 = -2.99m1
Initial momentum of the second cart = m2 * v2 = 28.5 * 2.04

The total momentum before the collision is the sum of the individual momenta:
Total initial momentum = -2.99m1 + 28.5 * 2.04

After the collision, the first cart comes to rest, so its final velocity is 0 m/s. The second cart changes direction and moves to the left with the same speed of 2.04 m/s.

The momentum after the collision can be calculated as:
Final momentum of the first cart = 0 * m1 = 0
Final momentum of the second cart = -m2 * v2 = -28.5 * 2.04

The total momentum after the collision is the sum of the individual momenta:
Total final momentum = 0 + (-28.5 * 2.04)

According to the conservation of momentum, the total initial momentum should be equal to the total final momentum. Therefore, we can set up the equation:

-2.99m1 + 28.5 * 2.04 = 0 + (-28.5 * 2.04)

Simplifying this equation, we get:
-2.99m1 + 58.14 = -58.14

Now, we can solve for m1:
-2.99m1 = -116.28
m1 = -116.28 / -2.99

Calculating this, we find:
m1 ≈ 38.8 kg

Therefore, the mass of the first cart is approximately 38.8 kg.