A 68.0 kg person throws a 0.0450 kg snowball forward with a ground speed of 27.0 m/s. A second person, with a mass of 58.0 kg, catches the snowball. Both people are on skates. The first person is initially moving forward with a speed of 2.50 m/s, and the second person is initially at rest. What are the velocities of the two people after the snowball is exchanged? Disregard the friction between the skates and the ice.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the snowball is thrown is equal to the total momentum after the snowball is caught.

Let's assume the first person is person A and the second person is person B.

Given:
Mass of person A (m1) = 68.0 kg
Mass of person B (m2) = 58.0 kg
Mass of snowball (m3) = 0.0450 kg

Initial velocity of person A (u1) = 2.50 m/s
Initial velocity of person B (u2) = 0 m/s (at rest)
Velocity of snowball (u3) = 27.0 m/s (thrown forward)

To find the final velocities of person A (v1) and person B (v2), we'll use the conservation of momentum equation:

(m1 * u1) + (m3 * u3) = (m1 * v1) + (m2 * v2)

Now let's substitute the values:

(68.0 kg * 2.50 m/s) + (0.0450 kg * 27.0 m/s) = (68.0 kg * v1) + (58.0 kg * v2)

170.0 kg*m/s + 1.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2
171.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2

We also know that after the exchange of the snowball, the total momentum of the system should be conserved. Since momentum is a vector quantity, we can write this in terms of the magnitude of momentum:

(m1 * u1) + (m2 * u2) + (m3 * u3) = (m1 * v1) + (m2 * v2)

Now let's substitute the values:

(68.0 kg * 2.50 m/s) + (58.0 kg * 0 m/s) + (0.0450 kg * 27.0 m/s) = (68.0 kg * v1) + (58.0 kg * v2)

170.0 kg*m/s + 0 kg*m/s + 1.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2
171.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2

Now we have a system of equations:

171.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2
171.215 kg*m/s = 68.0 kg * v1 + 58.0 kg * v2

This can be solved simultaneously to find the values of v1 and v2.

To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the snowball is thrown is equal to the total momentum after the snowball is caught.

The momentum of an object is given by the product of its mass and velocity. So, let's calculate the momentum of each individual before and after the snowball exchange.

Let's take the positive direction to be the forward direction.

For the first person before the snowball exchange:
Mass of the first person, m1 = 68.0 kg
Initial velocity of the first person, u1 = 2.50 m/s

The momentum of the first person before the snowball exchange is given by:
Momentum of the first person before = m1 * u1

For the snowball:
Mass of the snowball, m_snowball = 0.0450 kg
Initial velocity of the snowball, u_snowball = 27.0 m/s

The momentum of the snowball before the exchange is given by:
Momentum of the snowball before = m_snowball * u_snowball

For the second person before the snowball exchange:
Mass of the second person, m2 = 58.0 kg
Initial velocity of the second person, u2 = 0 m/s (at rest)

The momentum of the second person before the snowball exchange is given by:
Momentum of the second person before = m2 * u2

According to the principle of conservation of momentum, the total initial momentum before the snowball exchange should be equal to the total final momentum after the snowball exchange.

Total initial momentum = Momentum of the first person before + Momentum of the snowball before + Momentum of the second person before

Let's calculate the total initial momentum.

Total initial momentum = (m1 * u1) + (m_snowball * u_snowball) + (m2 * u2)

Now, let's consider the velocities of the two people after the snowball exchange:

For the first person after the snowball exchange:
Mass of the first person, m1 = 68.0 kg
Final velocity of the first person, v1 (to be determined)

For the second person after the snowball exchange:
Mass of the second person, m2 = 58.0 kg
Final velocity of the second person, v2 (to be determined)

According to the principle of conservation of momentum, the total final momentum after the snowball exchange should be equal to the total initial momentum before the exchange.

Total final momentum = (m1 * v1) + (m_snowball * v_snowball) + (m2 * v2)

Since the snowball is caught by the second person and comes to rest, the final velocity of the snowball (v_snowball) is 0 m/s.

Total final momentum = (m1 * v1) + (m_snowball * 0) + (m2 * v2)

Now, let's equate the total initial momentum to the total final momentum and solve for v1 and v2:

Total initial momentum = Total final momentum

(m1 * u1) + (m_snowball * u_snowball) + (m2 * u2) = (m1 * v1) + (m_snowball * 0) + (m2 * v2)

Simplifying the equation:

(m1 * u1) + (m_snowball * u_snowball) + (m2 * u2) = (m1 * v1) + (m2 * v2)

Now we can substitute the given values and solve for v1 and v2.