A skater of mass 74.0 kg initially moves in a straight line at a speed of 4.60 m/s. The skater approaches a child of mass 45.0 kg, whom he lifts on his shoulders. Assuming there are no external horizontal forces, what is the skater's final velocity?

To find the skater's final velocity, we can apply the principle of conservation of linear 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 given by the product of its mass and its velocity:

Momentum = mass × velocity

Let's denote the skater's initial velocity as v1, the child's initial velocity as v2, and the final velocity of the system (the skater and the child together) as vf.

Momentum before interaction = Momentum after interaction

Total initial momentum = Total final momentum

The total initial momentum is the sum of the initial momenta of the skater and the child:

(mass of the skater × initial velocity of the skater) + (mass of the child × initial velocity of the child) = (mass of the skater + mass of the child) × final velocity

Plugging in the values:

(74.0 kg × 4.6 m/s) + (45.0 kg × 0 m/s) = (74.0 kg + 45.0 kg) × final velocity

(340.4 kg·m/s) = (119.0 kg) × final velocity

Now, divide both sides of the equation by the total mass of the system:

final velocity = (340.4 kg·m/s) / (119.0 kg)

final velocity ≈ 2.86 m/s

Therefore, the skater's final velocity, after lifting the child on his shoulders, is approximately 2.86 m/s.