A skater of mass 80.5 kg initially moves in a straight line at a speed of 4.90 m/s. The skater approaches a child of mass 41.0 kg, whom he lifts on his shoulders. Assuming there are no external horizontal forces, what is the skater's final velocity?
Momentum is conserved:
80.5*4.90=(80.5+41)*V solve for V
To find the skater's final velocity, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, as long as there are no external forces acting on the system.
The momentum of an object can be calculated by multiplying its mass by its velocity:
momentum = mass × velocity
Before the child is lifted onto the skater's shoulders, the skater and the child move separately with their respective velocities. Let's represent the velocity of the skater as v1 and the velocity of the child as v2.
The momentum of the skater before the child is lifted can be calculated as:
momentum1 = mass1 × v1
momentum1 = (80.5 kg) × (4.90 m/s)
momentum1 = 394.45 kg·m/s
The momentum of the child before being lifted is given by:
momentum2 = mass2 × v2
momentum2 = (41.0 kg) × (0 m/s) (since the child is initially at rest)
momentum2 = 0 kg·m/s
After the child is lifted onto the skater's shoulders, they both move together with a final velocity, which we'll represent as vf.
The total momentum after the child is lifted can be calculated by adding the individual momenta of the skater and the child:
momentum_total = momentum1 + momentum2
momentum_total = 394.45 kg·m/s + 0 kg·m/s
momentum_total = 394.45 kg·m/s
Now, to find the final velocity of the skater-child system (vf), we equate the total momentum before and after the child is lifted:
momentum_total = mass_total × vf
mass_total = mass1 + mass2 (sum of the masses)
mass_total = (80.5 kg) + (41.0 kg)
mass_total = 121.5 kg
Therefore:
momentum_total = (121.5 kg) × vf
vf = momentum_total / mass_total
vf = (394.45 kg·m/s) / (121.5 kg)
vf ≈ 3.246 m/s
Hence, the skater's final velocity, when lifting the child, is approximately 3.246 m/s.