A 74.0 kg ice skater moving to the right with a velocity of 2.58 m/s throws a 0.18 kg snowball to the right with a velocity of 24.4 m/s relative to the ground.
(a) What is the velocity of the ice skater after throwing the snowball? Disregard the friction between the skates and the ice.
(b) A second skater initially at rest with a mass of 60.00 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
To find the velocities of the ice skater and the second skater, we can use the principle of conservation of momentum. This principle states that the total momentum before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.
(a) What is the velocity of the ice skater after throwing the snowball?
Let's denote the initial velocity of the ice skater as v1 and the velocity of the snowball as v2. The mass of the ice skater is 74.0 kg, and the mass of the snowball is 0.18 kg.
Using the principle of conservation of momentum, we can write:
Total initial momentum = Total final momentum
(mass of ice skater * initial velocity of ice skater) + (mass of snowball * initial velocity of snowball)
= (mass of ice skater * final velocity of ice skater) + (mass of snowball * final velocity of snowball)
(74.0 kg * 2.58 m/s) + (0.18 kg * 24.4 m/s) = (74.0 kg * final velocity of ice skater) + (0.18 kg * 0)
Simplifying the equation:
191.32 kg·m/s + 4.392 kg·m/s = (74.0 kg * final velocity of ice skater)
Adding the two terms on the left side:
195.712 kg·m/s = (74.0 kg * final velocity of ice skater)
Dividing both sides of the equation by 74.0 kg:
final velocity of ice skater = 195.712 kg·m/s / 74.0 kg
final velocity of ice skater ≈ 2.643 m/s
Therefore, the velocity of the ice skater after throwing the snowball is approximately 2.643 m/s to the right.
(b) What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
In a perfectly inelastic collision, the two objects stick together and move as one after the collision. Let's denote the final velocity of the combined skaters as v3.
Using the principle of conservation of momentum, we can write:
Total initial momentum = Total final momentum
(mass of ice skater * final velocity of ice skater) + (mass of snowball * final velocity of snowball)
= (mass of combined skaters * final velocity of combined skaters)
(74.0 kg * 2.643 m/s) + (0.18 kg * 24.4 m/s) = (74.0 kg + 60.0 kg) * final velocity of combined skaters
(195.582 kg·m/s) = (134.0 kg) * final velocity of combined skaters
Dividing both sides of the equation by 134.0 kg:
final velocity of combined skaters = 195.582 kg·m/s / 134.0 kg
final velocity of combined skaters ≈ 1.458 m/s
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 1.458 m/s to the right.