A 45 kg boy is holding a 12 kg pumpkin while standing on ice skates on a smooth frozen pond. The boy tosses the pumpkin with a horizontal speed of 2.8 m/s towards a 37 kg girl who catches it. The girl is also wearing ice skates. What are the final speeds of the boy and the girl?

To find the final speeds of the boy and the girl, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the interaction is equal to the total momentum after the interaction.

The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by:

p = m * v

where p is momentum, m is mass, and v is velocity.

Given that the boy's mass (m1) is 45 kg, the pumpkin's mass (m2) is 12 kg, the boy's initial horizontal velocity (v1) is 0 m/s (since he is standing still), and the pumpkin's horizontal velocity (v2) is 2.8 m/s, we can calculate the total initial momentum.

Total initial momentum = m1 * v1 + m2 * v2

Total initial momentum = (45 kg * 0 m/s) + (12 kg * 2.8 m/s)

Total initial momentum = 0 kg*m/s + 33.6 kg*m/s

Total initial momentum = 33.6 kg*m/s

After the girl catches the pumpkin, both the boy and the girl move in the same direction with a common final velocity (v3). Assuming the final velocity of the boy is v1' and the final velocity of the girl is v3, we can rewrite the total momentum equation as:

Total final momentum = m1 * v1' + m2 * v3

Since the girl is also standing still initially (v1' = 0 m/s) and the total momentum is conserved, the equation becomes:

Total initial momentum = Total final momentum

m1 * v1 + m2 * v2 = m1 * v1' + m2 * v3

Substituting the given values:

(45 kg * 0 m/s) + (12 kg * 2.8 m/s) = (45 kg * v1') + (12 kg * v3)

0 kg*m/s + 33.6 kg*m/s = 45 kg * v1' + 12 kg * v3

33.6 kg*m/s = 45 kg * v1' + 12 kg * v3

We have two unknowns (v1' and v3), so we need another equation to solve for both velocities.

According to the conservation of mass, the total mass before the interaction is equal to the total mass after the interaction. Mathematically:

Total initial mass = Total final mass

m1 + m2 = m1 + m2

45 kg + 12 kg = 45 kg + 37 kg

57 kg = 82 kg

Since the total mass before and after the interaction is the same, we can conclude that the final velocity of the boy and the girl will depend only on the ratio of their initial masses (m1 and m2) and not on their individual masses.

Therefore, the final speeds of the boy and the girl will be the same and can be calculated using the equation:

Total initial momentum = Total final momentum

(45 kg * 0 m/s) + (12 kg * 2.8 m/s) = (45 kg + 12 kg) * v3

0 kg*m/s + 33.6 kg*m/s = 57 kg * v3

33.6 kg*m/s = 57 kg * v3

Therefore, the final speed of both the boy and the girl (v3) is:

v3 = 33.6 kg*m/s / 57 kg

v3 ≈ 0.589 m/s

So, the final speeds of both the boy and the girl are approximately 0.589 m/s.

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