A skater of mass 85.5 kg initially moves in a straight line at a speed of 4.70 m/s. The skater approaches a child of mass 36.5 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 after lifting the child, we can use the law of conservation of momentum.

The law of conservation of momentum states that the total momentum of a system before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.

The momentum of an object is defined as the product of its mass and velocity. In this case, we need to consider the initial and final momentum of the skater-child system.

The initial momentum of the skater is given by:
Initial momentum of the skater = mass of skater * initial velocity of skater

= 85.5 kg * 4.70 m/s

The initial momentum of the child is given by:
Initial momentum of the child = mass of child * initial velocity of child

= 36.5 kg * 0 m/s (since the child is stationary before being lifted)

After the skater lifts the child, they move together as one system. The final velocity of the skater-child system is the final velocity of the combined mass.

Let's denote the final velocity of the skater-child system as Vf.

The final momentum of the skater-child system is given by:
Final momentum of the skater-child system = (mass of skater + mass of child) * final velocity of skater-child system

= (85.5 kg + 36.5 kg) * Vf

According to the law of conservation of momentum, the initial momentum of the skater-child system is equal to the final momentum:

Initial momentum of the skater + Initial momentum of the child = Final momentum of the skater-child system

(85.5 kg * 4.70 m/s) + (36.5 kg * 0 m/s) = (85.5 kg + 36.5 kg) * Vf

Simplifying the equation, we get:

(85.5 kg * 4.70 m/s) = (85.5 kg + 36.5 kg) * Vf

Now we can solve for Vf:

Vf = (85.5 kg * 4.70 m/s) / (85.5 kg + 36.5 kg)

Vf = 400.85 kg·m/s / 122 kg

Vf ≈ 3.28 m/s

Therefore, the skater's final velocity after lifting the child is approximately 3.28 m/s.

To find the skater's final velocity, we can use the principle of conservation of momentum. The total momentum before the child is lifted is equal to the total momentum after the child is lifted.

The momentum of an object is defined as the product of its mass and velocity. Mathematically, it can be represented as:

Momentum (p) = mass (m) × velocity (v)

Before the child is lifted, the skater's momentum can be calculated as:

Momentum of the skater before = mass of the skater × velocity of the skater

= 85.5 kg × 4.70 m/s

The momentum of the skater and child after the child is lifted can be calculated as:

Momentum of the skater and child after = (mass of the skater + mass of the child) × final velocity

= (85.5 kg + 36.5 kg) × final velocity

Since the total momentum before is equal to the total momentum after, we can set up the following equation:

Momentum of the skater before = Momentum of the skater and child after

85.5 kg × 4.70 m/s = (85.5 kg + 36.5 kg) × final velocity

Simplifying this equation, we have:

402.85 kg·m/s = 122.0 kg × final velocity

To calculate the final velocity, divide both sides of the equation by 122.0 kg:

final velocity = (402.85 kg·m/s) / 122.0 kg

final velocity ≈ 3.30 m/s

Therefore, the skater's final velocity is approximately 3.30 m/s.