A 74.0 kg ice skater moving to the right with a velocity of 2.14 m/s throws a 0.12 kg snowball to the right with a velocity of 24.2 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 60.50 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?
(a) To determine the velocity of the ice skater after throwing the snowball, we can use the principle of conservation of linear momentum. The initial momentum of the ice skater and the snowball is equal to the final momentum of the skater alone. Mathematically, this can be written as:
(m1 * v1) + (m2 * v2) = (m1 * vf)
Where:
m1 = mass of the ice skater = 74.0 kg
v1 = initial velocity of the ice skater = 2.14 m/s
m2 = mass of the snowball = 0.12 kg
v2 = initial velocity of the snowball = 24.2 m/s
vf = final velocity of the ice skater
Substituting the given values into the equation:
(74.0 kg * 2.14 m/s) + (0.12 kg * 24.2 m/s) = (74.0 kg * vf)
We can solve for vf:
(158.36 kg·m/s) + (2.904 kg·m/s) = (74.0 kg * vf)
161.264 kg·m/s = 74.0 kg * vf
Dividing both sides by 74.0 kg:
2.18373 m/s = vf
Therefore, the velocity of the ice skater after throwing the snowball is approximately 2.18373 m/s to the right.
(b) In a perfectly inelastic collision, the two skaters stick together and move with a common final velocity. We can again use the principle of conservation of linear momentum to determine this velocity. The initial momentum of the snowball is equal to the final momentum of the two skaters. Mathematically, this can be written as:
(m2 * v2) + (m3 * v3) = (m2 + m3) * vfinal
Where:
m2 = mass of the snowball = 0.12 kg
v2 = initial velocity of the snowball = 24.2 m/s
m3 = mass of the second skater = 60.50 kg
v3 = initial velocity of the second skater = 0 m/s (initially at rest)
vfinal = final velocity of the two skaters after the collision
Substituting the given values into the equation:
(0.12 kg * 24.2 m/s) + (60.50 kg * 0 m/s) = (0.12 kg + 60.50 kg) * vfinal
(2.904 kg·m/s) + (0 kg·m/s) = (60.62 kg) * vfinal
2.904 kg·m/s = 60.62 kg * vfinal
Dividing both sides by 60.62 kg:
0.04782 m/s = vfinal
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 0.04782 m/s to the right.
(a) To find the velocity of the ice skater after throwing the snowball, we can use the principle of conservation of momentum. The total momentum before the snowball is thrown is equal to the total momentum after the snowball is thrown.
The initial momentum of the ice skater can be calculated as the product of their mass (m1) and initial velocity (v1).
Initial momentum of ice skater (m1v1) = (74.0 kg)(2.14 m/s) = 158.36 kg⋅m/s
The momentum of the snowball can be calculated as the product of its mass (m2) and velocity (v2).
Momentum of snowball (m2v2) = (0.12 kg)(24.2 m/s) = 2.904 kg⋅m/s
Since momentum is conserved, the total momentum after the snowball is thrown is equal to the sum of the initial momentum of the ice skater and the momentum of the snowball.
Total momentum after snowball is thrown = 158.36 kg⋅m/s + 2.904 kg⋅m/s = 161.264 kg⋅m/s
The final velocity of the ice skater (v) can be found by dividing the total momentum after the snowball is thrown by the mass of the ice skater.
Final velocity of ice skater (v) = Total momentum after snowball is thrown / Mass of ice skater
v = 161.264 kg⋅m/s / 74.0 kg
v ≈ 2.18 m/s
Therefore, the velocity of the ice skater after throwing the snowball is approximately 2.18 m/s to the right.
(b) In a perfectly inelastic collision, the two objects stick together after the collision and move with a common velocity.
To find the velocity of the second skater after catching the snowball, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The initial momentum of the second skater is zero since they are initially at rest.
Initial momentum of second skater = 0 kg⋅m/s
The momentum of the snowball is the same as calculated earlier: 2.904 kg⋅m/s.
Since momentum is conserved, the total momentum after the collision is equal to the sum of the initial momentum of the second skater and the momentum of the snowball.
Total momentum after collision = 0 kg⋅m/s + 2.904 kg⋅m/s = 2.904 kg⋅m/s
The final velocity of the second skater (v') can be found by dividing the total momentum after the collision by the combined mass of the second skater and the snowball.
Combined mass of second skater and snowball = Mass of second skater + Mass of snowball
= 60.50 kg + 0.12 kg
= 60.62 kg
Final velocity of second skater (v') = Total momentum after collision / Combined mass of second skater and snowball
v' = 2.904 kg⋅m/s / 60.62 kg
v' ≈ 0.048 m/s
Therefore, the velocity of the second skater after catching the snowball in a perfectly inelastic collision is approximately 0.048 m/s to the right.